sin

His main sin appears to be his poor timing.

12454It is not a sin to want your child.

7241Sin is the contradiction of that purpose, and guilt is alienation from the family.

2613"Ah, what have you done to me?" it still seemed to say, and Prince Andrew felt that something gave way in his soul and that he was guilty of a sin he could neither remedy nor forget.

1917I've been so focused on sin that I have been...

1918Ogling a spouse couldn't be a sin – especially when he derived such obvious pleasure from it.

128Powers created by worldliness and sin are crumbling, as they well may; "the city of God remaineth!"

1210The fact that sin exists, and that great misery results from it, dawned gradually upon her mind as she understood more and more clearly the lives and experiences of those around her.

1012Call it a sin of omission.

1013It's a sin... outside marriage.

914She has done this deed for so many others, but I can't bear to heap more sin on my blackened soul and kill unborn this result of my Joshua's love.

89Several indications favour the view of the connexion in the age of Moses between the Yahweh-cult at Sinai and the moon-worship of Babylonian origin to which the name Sinai points (Sin being the Babylonian moon-god).

67After the first blush of sin comes its indifference; and from immoral it becomes, as it were, unmoral, and not quite unnecessary to that life which we have made.

55If now you married again with the object of bearing children, your sin might be forgiven.

55He is appointed watchman to warn men when they sin, and is to be held responsible for the consequences if he fails in this duty.

56" If we clearly see that what we are doing is wrong, it would be impossible for us to sin, so long as we saw it in that light."

44That is why it is a sin for men like you, Prince, not to serve in these times!

44"The sin once committed, there is no more wavering or flinching possible to him, who has fought so hard against the demoniac possession; while she who resigned body and soul to the tempter, almost at a word, remains liable to the influences of religion and remorse."

45This element of public confession for sin became more prominent in the days when synagogal worship developed, and prayer took the place of the sacrificial offerings which could only be offered in the Jerusalem temple.

45So Josephus saved them from the sin of suicide and gave himself up to the Romans.

45It's a sin to speak so.

46It is, of course, true that the ethical conception of sin as violation of righteousness and an act of rebellion against the divine righteous will had been developed since the days of Amos and Isaiah; but, as we have already observed, cultus and prophetic teaching were separated by an immense gulf, and in spite of the reformation of 621 B.C. still remain separated.

32The enzymes which act upon glucosides are many; the best known are emul sin and myrosin, which split up respectively amygdalin, the special glucoside of certain plants of the Rosaceae; and sinfgrin, which has a wide distribution among those of the Cruciferae.

33A venial sin, for you acted without evil intention.

33The icicles are prison bars on our windows, trapping us, prisoners to this life of sin and degradation...

34Imagine the overwhelming guilt Rev. Martin must have felt over this terrible sin of his relationship with a prostitute.

34The belief was taught in the homogeneity of all living things, in the doctrine of original sin, in the transmigration of souls, in the view that the soul is entombed in the body (v13µa ojia), and that it may gradually attain perfection during connexion with a series of bodies.

34We are here moving in a realm of ideas prevailing in ancient Israel respecting holiness, uncleanness and sin, which are ceremonial and not ethical; see especially Robertson Smith's Religion of the Semites, 2nd ed., p. 446 foll.

39the sale of tithes, the taking of a fee for confession, absolution, marriage or burial, the concealment of one in mortal sin or the reconcilement of an impenitent for the sake of gain, and the doing homage for spiritualities.

20Tennant's Origin and Propagation of Sin (1902) - sin a " bye-product " of a generally good evolution.

22Randy has been so good about it—so encouraging, but even so—I feel like it's a sin to smile, or laugh.

23Once more, in the doctrine of sin and redemption, the governing idea is God's fatherly purpose for His family.

23The family name of the principal branch of this tribe is Abu Sin, and Gedaref, an important town in the centre of the Shukria country, was formerly called Suk Abu Sin.

231 Calvin suggested that men of known worth should be appointed in different quarters of the city to report to the ministers those persons in their district who lived in open sin; that the ministers should then warn such persons not to come to the communion; and that, if their warnings were unheeded, discipline should be enforced.

23In the tenth book of the Republic we find the curious argument that the soul does not perish like the body, because its characteristic evil, sin or wickedness does not kill it as the diseases of the body wear out the bodily life.

23Lowe was delighted with this, and promptly translated it into Latin, as follows: "Centinentur hac in fossa Humilis Roberti ossa; Si ad coelum evolabit, Pax in coelo non restabit; Sin in inferis jacebit, Diabolum ejus poenitebit."

23SIN, the name of the moon-god in Babylonia and Assyria, also known as Nannar, the "illuminer."

23After the defeat of the Abyssinians at Debra Sin in August 1887 Gondar was looted and fired by the dervishes under Abu Anga.

23The factors Af (u-v cos i) and Bf (v sin i) give the frictional resistance to sinking, per unit length of the cable, in the direction of the length and transverse to the length respectively.

23It was a noble end to what, in spite of its besetting sin of infirmity of moral purpose, was a not ignoble life.

23Pain and sin must have been reduced to a minimum by God; though they are so ingrained in the finite that we have to make up our minds even to the endless sin and endless punishments of hell.

23The confessions of sin which he introduced descend to minute ritual details and rise to the most exalted aspects of social and spiritual life.

23"I often think, though, perhaps it's a sin," said the princess, "that here lives Count Cyril Vladimirovich Bezukhov so rich, all alone... that tremendous fortune... and what is his life worth?

23I'd drench him in sin and guilt.

24"It sounds like sin was a very big business," Cynthia offered as she stirred a pot of fish chowder.

24God can dumbness keep While Sin creeps grinning through His house of Time.

24Yes; if I have a sin, a great sin, it is hatred of that vile woman! almost shrieked the princess, now quite changed.

24At this point, abandoning the two fertilized eggs might be a worse sin.

10She knew very little of Hannah's friends, except they were all richer than sin.

10(2) The acceleration of the element at the origin is - n 2 sin nt; so that the force which would have to be applied to the parts where the density is D' (instead of D), in order that the waves might pass on undisturbed, is, per unit of volume, (D' - D)n 2 sin nt.

10The element of area being 22rr 2 sin 04,, we have f:2 l 2x r2 si n 2 d ?=gam, r so that the energy emitted from T is represented by 87r3 (D, - D) 2 T2 (9) D2 x4' on such a scale that the energy of the primary wave is unity per unit of wave-front area.

10The name of Sin's chief sanctuary at Ur was E-gish-shir-gal, "house of the great light"; that at Harran was known as E-khul-khul, "house of joys."

10Or it is the doctrine of unfallen man's " natural state " - a doctrine intensified in Protestantism - separating itself from the theologians' grave doctrine of sin.

10Some of the Jews had married women of Ashdod, Ammon and Moab, and the impetuous governor indignantly adjured them to desist from a practice which was the historic cause of national sin.

10(2) Hebrew has one more sibilant than Arabic or Syriac: thus, as corresponding to s (samekh), s (sin) sh in Hebrew, Arabic has only s (sin) sh, while Syriac has a different pair s (samekh) sh.

10sin and Syr.

10sin (Syr.

10The sense of sin can hardly be said to enter into these exercises - that is, they are not undertaken as penance for personal transgression.

10(Porphyry tells us that Plotinus was unwilling to name his parents or his birthplace, and seemed ashamed of being in the body.) Beyond the uaOap ra, or virtues which purify from sin, lies the further stage of complete identification with God (ovrc w aµaprias Eivac; aXAa 0E6v Elm).

10His great sin in the matter of Uriah would have been forgotten but for his repentance: the things at which modern ideas are most offended are not always those that would have given umbrage to early writers.

10From this follows the necessity for the created spirit, after apostasy, error and sin, to return always to its origin in God.

10Of the names of the planets Estera (Ishtar Venus, also called Ruha d'Qudsha, "holy spirit"), Enba (Nebo, Mercury), Sin (moon), Kewan (Saturn), Bil (Jupiter), and Nirig (Nirgal, Mars) reveal their Babylonian origin; Il or Il Il, the sun, is also known as Kadush and Adunay (the Adonai of the Old Testament); as lord of the planetary spirits his place is in the midst of them; they are the source of all temptation and evil amongst men.

10The elliptic lemniscate has for its equation (x 2 +31 2) 2 =a 2 x 2 +b 2 y 2 or r 2 = a 2 cos 2 9 +b 2 sin 20 (a> b).

10The hyperbolic lemniscate has for its equation (x2 +y2)2 = a2x2 - b 2 y 2 or r 2 =a 2 cos 2 0 - b 2 sin 2 B.

10On the problem of evil and sin it is impossible here to enter; but this must be insisted on, that the miracles of Jesus at least express divine benevolence just under those conditions in which the course of nature obscures it, and are therefore, proper elements in a revelation of grace, of which nature cannot give any evidence.

10In the southwest of Manchuria a line of the imperial railways of Northern China gives connexion from Peking, and branches at Kou-pang-tsze to Sin Population.

10If G is the acceleration of gravity at the equator and g that at any latitude X, then g= G(IFoï¿½o0513 sin 2 X).

1020, no man is free from sin; vii.

10Under the new settlement Athens remained a free and sovereign city - a boon which she repaid by zealous Caesar-worship, for the favours bestowed upon her tended to pauperize her citizens and to foster their besetting sin of calculating flattery.

10Its object is to convince a man of sin, of justice and of judgment.

10The first week is the foundation, and has to do with the consideration of the end of man, sin, death, judgment and hell.

10Having purified the soul from sin and obtained a detestation thereof, the second week treats of the kingdom of Christ, and is meant to lead the soul to make an election of the service of God.

10If he says that a subject is to allow himself to be moved and directed, under God, by a superior just as though he were a corpse or as a staff in the hands of an old man, he is also careful to say that the obedience is only due in all things "wherein it cannot be defined (as it is said) that any kind of sin appears."

10It was thought that martyrdom would atone for sin, and imprisoned confessors not only issued to the Churches commands which were regarded almost as inspired utterances, but granted pardons in rash profusion to those who had been excommunicated by the regular clergy, a practice which caused Cyprian and his fellow bishops much difficulty.

10Then we have sin 2 D =sin a sin zt, and since sin a=sin (90°-1) = cos 1, it follows that sin ID = cos 1 sin it.

10Thus, cos D - cos a cos b cos sin a sin b cos D = cos a cos b + sin a sin b cos t = sin 1 sin l' + cos 1 cos l cos t.

10Contrary to the Puritan teaching of the time, they insisted on the possibility, in this life, of complete victory over sin.

10The writer already sees the Messianic kingdom established, under the sway of which the Gentiles will in due course be saved, Beliar overthrown, sin disappear from the earth, and the righteous dead rise to share fr1 the blessedness of the living.

10Up to this time Wesley says he had no notion of inward holiness, but went on "habitually and for the most part very contentedly in some or other known sin, indeed with some intermission and short struggles especially before and after Holy Communion," which he was obliged to attend three times a year.

10In 1765 he said it "contains all that I now teach concerning salvation from all sin, and loving God with an undivided heart."

10Victory over sin was the goal which he set before all his people.

10There shall be no more sin, no more temptation, no more suffering.

10They do not attempt a psychological explanation of the origin of human sin; bad thought (yeser ra`, Ecclus.

10Reform of this system, and, further, very necessary reforms of the methods of collection of the wines and spirits revenue (which is protection turned upside down, the home-growers being far more heavily taxed than importers), and of the customs (in which almost every possible administrative sin was exemplified), were also undertaken.

10But he showed admirable judgment in his choice of subordinates; Robert of Meulan, who died in 1118, and Roger of Salisbury, who survived his master, were statesmen of no common order; and Henry was free from the mania of attending in person to every detail, which was the besetting sin of medieval sovereigns.

10His system declared that holiness and sin are free voluntary exercises; that men act freely under the divine agency; that the slightest transgression deserves eternal punishment; that it is through God's mere grace that the penitent believer is pardoned and justified; that, in spite of total depravity, sinners ought to repent; and that regeneration is active, not passive, with the believer.

10The sacrament of the most Holy Eucharist shall be freely administered in the two kinds, that is bread and wine, to all the faithful in Christ who are not precluded by mortal sin - according to the word and disposition of Our Saviour.

10Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

10Of the quadratic axe+2bxy+cy2, he discovered the two invariants ac-b 2, a-2b cos w+c, and it may be verified that, if the transformed of the quadratic be AX2=2BXY+CY2, sin w 2 AC -B 2 =) (ac-b2), sin w A-2B cos w'+C = (sin w'1 2(a - 2bcosw+c).

10sin w The fundamental fact that he discovered was the invariance of 2 COS w xy+y 2, viz.

10We have cos w' = cos w = o and the substitution x 1 =cos OX, -sin 0(2 x 2 = sin OX i +cos 6X2, with modulus unity.

10If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin °Xicos OX2r of modulus -1.

10In both cases ddl and dal are cogredient with xl and x 2; for, in the case of direct substitution, dxi = cost dX i - sin 00-(2, ad2 =sin B dX i +cos O dX 2, and for skew substitution dai = cos B dX i +sin 0d2, c-&-- 2 n d =sin -coseax2.

10(aixi+a2x2)"`=4, and the direct stitution xi = XXi -, LX2, X2 =, hX replacing cos 0, sin 0 respectively.

10These were huge digests of all that popes, councils, primitive fathers had decided on every kind of question pertaining to the confessional - what exactly is a sin, what kind of questions the priests must ask, under what conditions he could give absolution.

10This made it a grave sin in the priest to refuse absolution, whenever there was some good reason for giving it even when there were other and better reasons for refusing it.

103) by F, and its components parallel to the co-ordinate axes by X and Y, we have X= - ax = M(3 cos' 0 - I), Y= - y = M (3 sin 0 cos 0.

10If F T is the force along r and F t that along t at right angles to r, F r =X cos 0+ Y sin 0=M 2 cos 0, F t = - X sin 0+ Y cos 0 = - r 3 sin 0.

10If a small magnet of moment M is placed in the sensibly uniform field H due to a distant magnet, the couple tending to turn the small magnet upon an axis at right angles to the magnet and to the force is MH sin 0, (17) where 0 is the angle between the axis of the magnet and the direction of the force.

10It can be deduced from (17), (12) and (13) that the couple on S'N' due to SN, and tending to increase 4), is MM' (sin 0 cos 4-2 sin 4) cos 0)/r'.

10(18) This vanishes if sin 0 cos 4)=2 sin 4 cos 0, i.e.

10The components X, Y, parallel and perpendicular to r, of the force between the two magnets SN and S'N' are X =3MM'(sin 0 sin 4)-2 cos 0 cos 4)/r 4, (21) Y=3MM'(sin 0 cos 4-{-sin 4 cos 0)/r 4 .

10If P is the weight of the magnet, l the length of each of the two threads, 2a the distance between their upper points of attachment, and 2b that between the lower points, then, approximately, MH = P(ab/l) sin 0.

10Let 0 be the angle which the standard magnet M makes with the meridian, then M'/R = sin 0, and M/R = cos 0, whence M' = M tan 0.

10He presupposes a nation of Yahweh-worshippers, whose religion has its centre in the temple and priesthood of Zion, which is indeed conscious of sin, and needs forgiveness and an outpouring of the Spirit, but is not visibly divided, as the kingdom of Judah was.

10The impressiveness and the stimulating power of the mystic ceremonies, the consciousness of being the privileged possessor of the secret wisdom of the ancients, the sense of purification from sin, and the expectation of a better life where there was to be compensation for the sufferings of this world - were all strong appeals to human nature.

10From that time Conselheiro was a victim of remorse, and to expiate his sin became a missionary in the sertao or interior of Brazil among the wild Jagunco people.

10If the primary wave at 0 be cos kat, the effect of the secondary wave proceeding from the element dS at Q is dS 1 dS - p cos k(at - p+ 4 A) = - -- sin k(at - p).

10" I am white," not " I am equal to white," or in the relative, I am born in sin," not " I am equal to born in sin."

10The directed line whose length is a, and which makes an angle 0 with the real (positive) unit line, is expressed by a (cos 0+i sin e), where i is regarded as +1,/ - 1.

10The product, P, of two such lines is, as we have seen, given by 1: a (cos B + i sin B):: a' (cos B' + i sin 0'): P, or P=aa' {cos (0+0')+i sin (0-{-0')}.

10ee and Argand had done, the expression a(cos 8 + i sin 0) as a directed line, let us suppose it to represent the operator which, when applied to any line in the plane in which 0 is measured, turns it in that plane through the angle 0, and at the same time increases its length in the ratio a : 1.

10purgatorium, from purgare, to purge), according to Roman Catholic faith, a state of suffering after death in which the souls of those who die in venial sin, and of those who still owe some debt of temporal punishment for mortal sin, are rendered fit to enter heaven.

10It appears, moreover, that if 0 be varied P will be least when L1H is at right angles to KLi, in which case Pi =W sin (aX), corresponding to 0 = X.

10If, as is usually most convenient, the two assigned directions are at right angles, the two components of a force P will be P cos 0, P sin 0, where 0 is the inclination of P to the direction of the p former component.

10Y = P1 sin Oi + P2 sin 02 + -..

10= ~ (P sin 0).

10If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are X, u, and if the axes are simultaneously turned through an angle e, the coordinates of a point of the lamina, relative to the original axes, are changed from x, y to X+x cos ey sin e, u+x sin e+y cos e, or X + x ye, u + Xe + y, ultimately.

10If the forces are all parallel, making say an angle 0 with Ox, we may write Xi = Pi cos 0, Vi = P1 sin 0, Xi = P2 cos 0, VI = Pi sin 0,.

10The equation (9) then becomes ~(xP) 1/2.~(P)) sin 8 {~(yP) i~.~(P)) cos 0=0.

10The displacement of the point C of the body is made up of 10 tangential to the meridian ZC and sin 0 ~,1 perpendicular to the plane of this meridian.

10sin a.

10sin ~9.

10sin (a+~).

10The displacement of the point H in the figure, resolved in the direction of R, is r cos 0sh sin 0.

10The work is therefore R(r cos 0sh sin 0)+ G cos 8

10=R4(p+p) cos 0h sin 0!, (3) if G=pR, r=pf, i.e.

10The factor (P+P) cos 0h sin 0 is called the vIrtual coefficient of the two screws which define the types of the wrench and twist, respectively.

10P we have (T + T) sin ai,L, or T4~, or Ts/p, where p is the radius of curvature.

10Hence OT=wOs sin ~, TO~=wOs cos ~+R&s.

10In particular, if we consider a length AP beginning at the lowest point A, then resolving horizontally and vertically we have T cos ~=T0, T sin ~ (6)

10If u be the acceleration at unit distance, the equation of motion becomes the solution of which may be written in either of the forms x=A cos et+B sin ut, x=a cos (rt+~), (6)

10The solution is x=A cos nl+B sinai, y=C cos nt-ED sin at, (9)

10If P be the initial position of the particle, we may conveniently take OP as axis of x, and draw Oy parallel to the direction of motion at P. If OP=a, and ~ be the velocity at P, we have, initially, x=a, y=o, x=o, y=.f0 whence x=a cos at, y=b sin nt, (10)

10dy my mv~= mg sin = mg-~, ~--= mg cos ~+R.

10The tangential resolution then gives Ia~1~1-= g sin ~.

10If the pendulum oscillate between the limits ~ = ~ a, we have (0~)2~g (cos ~cos a)=~f(sini 4asin23/4#); (1~) and, putting sin 3/4~ = sin 1/2a.

10sin 4, we find for the period (r) of a complete oscillation r= 4f ~4=~V- f1/8~r~ ~1 _Ii~in1~

10Fi(sin 1/811), (18)

10The function Fi (sin ~) was tabulated by A.

10This is satisfied by x=A cos (~t+), y=B sin (rt+e), (36) provided (g+(~2P)A+2~B =o,1 (

10Again, in the equiangular spiral we have p =r sin a, and therefore P = u/ri, if u =hh/sinh a.

10M~=Mg sin aF, o=Mg cos aR, MsiU=Fa (10)

10Thus the centre of a sphere rolling under gravity on a plane of inclination a describes a parabola with an acceleration g sin a/(I+C/Ma)

10The solution of these equations is of the type x = b cos (at+~), y=b sin (ot+), (28) where b, ~ are arbitrary, and 7a/c 2

10The motion of G then consists of two superposed circular vibrations of the type x=a cos (ai+s), ya sin (at+~), (36)

10If a be the inclination of the instantaneous axis to the axis of symmetry, (3 the inclination of the latter axis to the invariable line, we have rcosfl=Cw cos a, r sin ~3 = Aw sin a, (6) whence tan ~ =~ tan a.

10Also if ~ be the rate at which J describes the polhode, we have ~l sin (J3a) ~ sin (3, whence sin (aa)

10If the axis of the flywheel make an angle U with the vertical, it is seen in like manner that th~ required couple in the vertical plane through the axis is Cn sin 0 ~.

10The second governorship of Clive was marked by the transfer of the diwani or financial administration from the Mogul emperor to the Company, and by the enforcement of stringent regulations against the besetting sin of peculation.

13If dS =27rxdx, we have for the whole effect 27r œ sin k(at - p)x dx, f P ' or, since xdx = pdp, k = 27r/A, - k fr' sin k(at - p)dp= [- cos k(at - p)]°° r.

00The amplitude of the light at any point in the axis, when plane waves are incident perpendicularly upon an annular aperture, is, as above, cos k(at-r 1)-cos k(at-r 2) =2 sin kat sin k(r1-r2), r2, r i being the distances of the outer and inner boundaries from the point in question.

00The second and third factors of (3) being each of the form sin 2u/u2, we have to examine the character of this function.

00The maxima occur when u=tan u, (4), and then sin 2 u/u 2 = cos 2 u (5).

0015, p. 315) to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula isin 2 u f sin u du = du =7-.

00In the direction (suppose horizontal) for which n=o, /f=sin 0, the phases of the secondary waves range over a complete period when sin 0 =X/a, and, since all parts of the horizontal aperture are equally effective, there is in this direction a complete compensation and consequent absence of illumination.

00When sin 0 = 2A/a, the phases range one and a half periods, and there is revival of illumination.

00In like manner we may find the illumination in any other direction, and it is obvious that it vanishes when sin 0 is any multiple of A/a.

003), representing the value of the function sin 2 u/u 2 from u=o to u=271.

00(2), where S = ff sin(px+gy)dx dy,.

00This integral is the Bessel's function of order unity, defined by J,(z) n (z cos 0) sin 24 d4)..

00Trans., 1834) in his original investigation of the diffraction of a circular object-glass, and readily obtained from (6), is z z 3 25 27 J1(z) = 2 2 2.4 + 2 2.4 2.6 2 2.4 2.6 2.8 + When z is great, we may employ the semi-convergent series Ji(s) = A/ (7, .- z)sin (z-17r) 1+3 8 1 ' 6 (z) 2 3.5.7.9.1.3.5 5 () 3 1 3.5.7.1 1 3 cos(z - ?r) 8 ' z (z) 3.5.7.9.11.1.3.5.7 1 5 + 8.16.24.32.40 (z

00Thus, if x = p cos 4), y= p sin 0, C =11 cos px dx dy =f o rt 2 ' T cos (pp cos 0) pdp do.

00Now since AP is very small, AL' - PL'= AP sin a, where a is the angular semi-aperture L'AB.

00In like manner PL - AL has the same value, so that PL - PL' = 2AP sin a.

00Calling the refractive index µ, we have as the critical value of e=2Xo/ µ sin a, (1).

00The denominator sin a is the quantity well known (after Abbe) as the " numerical aperture."

00If 2R be the diameter of the objectglass and D the distance of the object, the angle subtended by AP is E/D, and the angular resolving power is given by X/2 D sin a = X/2 R (3) This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.

00(7) If B m denote the brightness of the mth lateral image, and Bo that the central image, we have amp 'cosx' dx= a d (1) (-) m7r B.: Bo= a+d am?r sin' a4 d (1).

00hen d =a the general formula becomes sin' Zm7r Bm: B = (3), showing that, when m is even, B m vanishes, and that, when m is odd, B m: B =1/m272.

006), and the diffracted rays make an angle ¢ (upon the same side), the relative retardation from each element of width (a+d) to the next is (a+d) (sin 9 +sin op); and this is the quantity which is to be equated to mX.

00Thus sin e +sin 0=2 sin 2(0+x) cos 2(0-0) = mX/(a +d) (5).

00To find the former, we have, if OAQ=4), AOP=w, QP 2 =u 2 +4a 2 sin 2 2w - 4au sin la) sin (2w-4)) = (u +a sin 4) sin w) 2 -a 2 sin 2 4)sin 2 c0+4a sin 2 2w(a-u cos 0).

00But if we now suppose that Q lies on the circle u= a cos 0, the middle term vanishes, and we get, correct as far as w4, QP= (u+a sin 4) sin w) 1 ' 3 1 {- a sin2c?sin4w V 4u so that QP - u=asin0sinw -Ft asin¢tanOsin 4 w..

00(9), in which it is to be noticed that the adjustment necessary to secure the disappearance of sin 2w is sufficient also to destroy the term in sin' w.

00A similar expression can be found for Q'P - Q"A; and thus, if Q' A =v, Q' AO = where v =a cos (0", we get - - -AQ' = a sin w (sin 4 -sink") - - 8a sin 4 w(sin cktan 4 + sin 'tan cl)').

00In the present application 4' is not necessarily equal to; but if P correspond to a line upon the grating, the difference of retardations for consecutive positions of P, so far as expressed by the term of the first order, will be equal to mX (m integral), and therefore without influence, provided v (sin 0-sin0') = nzX (11), where a denotes the constant interval between the planes containing the lines.

00For this purpose Rowland places the eye-piece at 0, so that 0 =o, and then by (11) the value of '" in the m th spectrum is o- sin $' = tmX.

00If w now relate to the edge of the grating, on which there are altogether n lines, no- = 2a sin w, and the value of the last term in (I o) becomes no- sin 3w sin O'tan 0', - 1 1 - 6 mnX sin' w tan 0'.

00f + 1 sin k a - f+ " dxdy = - 2 1h sin n k ?.

00If we put for shortness 7 for the quantity under the last circular function in (I), the expressions (i), (2) may be put under the forms u sin T, v sin (T - a) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin T and cos T in the expression u sin T +v sin (T - a), so that I =u 2 +v 2 +2uv cos a, which becomes on putting for u, v, and a their values, and putting f =Q .

00The constant multiplier is of no especial interest so that we may take as applicable to the image of a line 0 I = z 2 sin e A f 1+cos ` - 271 - Eh).

00In the second term if we observe that cos {p'+ 27rh/Af)E} =cos{p' - g,E} = cos p cos g, +sin p sin giE, we see that the second part vanishes when integrated, and that the remaining integral is of the form w = f +.0 sin z h, cos where h,=7rh/Af, g,=a-27Th/Af.

00v sin ?i, where a is the grating-interval and 43, the obliquity, the closeness of the grouping increasing with the number of intervals.

00By separation of real and imaginary parts, C =M cos 27rv 2 +N sin 27rv2 1 S =M sin 27rv 2 - N cos 27rv2 where 35+357.9 N _ 7rv 3 7r 3 v 7 + 1.3 1.3.5.7 1.3.5.7.9.11 These series are convergent for all values of v, but are practically useful only when v is small .

00dx ru i-x) C-Fi S= „ 7 o?I x o 1 fGO dx eu(ti-x) - 1 2Jo x i - x Thus, if we take _ 1 `°el 1 ('°° e uxdx G 7r12 Jo 1+ x 2 ' H 7r-N/2Jo -Vx.(1-i-x2)' C = 2-G cos u+ H sin u, S =1---G sin u-H cos u.

00(19), 1 abA) ' ' we may write 12= (cos 27rv 2 .dv) 2 + (f sin zirv 2 .dv) 2 (20), or, according to our previous notation, 12 = (2 - C 2 +(z - Sv)2= G2 +H2 Now in the integrals represented by G and H every element diminishes as V increases from zero.

00The intensity may be expressed by 12= (2+Cv) 2 +(2+Sv) 2 and the maxima and minima occur when dC dS (z+Cv)a`j+(2+Sv)dV=0, whence sin rV 2 +cos27rV 2 =G..

0019, where, according to the definition (5) of C, S, x =i v cos 27rv 2 .dv, y = f v sin ?7rv 2 .dv..

00For dx=cos airv 2 .dv, dy= sin 271-v2.dv; so that s = f (dx 2 +dy 2) =v, (30), 0= tan1 (dyldx) =171-v 2 (31).

00Then the displacement at 0 will take place in a direction perpendicular to 0 1 0, and lying in the plane Z0 1 0; and, if 1' be the displacement at 0, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive, = 4?y (1 +cos 0) sin 4 f' (bt - r).

00f(bt - x) =c sin 2 i n: (bt - x).

00we shall have '2y (1 +cos e)sin cos 2?

00The occurrence of sin 4 as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the direction B depend upon the condition of the former as regards polarization.

00Its connexion with a is expressed by a =c4'/dr; so that TZ sin 05 e'(at - kr) 47b 2 where the factor e int is restored.

00Retaining only the real part of (16), we find, as the result of a local application of force equal to DTZ cos nt (17), the disturbance expressed by TZ sin 4, cos(nt - kr) ?

00- 47rb 2 ' The occurrence of sin 4 shows that there is no disturbance radiated in the direction of the force, a feature which might have been anticipated from considerations of symmetry.

00We will now apply (18) to the investigation of a law of secondary disturbance, when a primary wave = sin (nt - kx) (19) is supposed to be broken up in passing the plane x = o.

00According to (18), the effect of the force acting at dS parallel to OZ, and of amount equal to 2b2kD dS cos nt, will be a disturbance - dS sin cos (nt - kr) (20), regard being had to (12).

00The proportionality of the secondary disturbance to sin 43 is common to the present law and to that given by Stokes, but here there is no dependence upon the angle 0 between the primary and secondary rays.

00The occurrence of factors such as sin 4), or 2 (1cos 0), in the expression of the secondary wave has no influence upon the result of the integration, the effects of all the elements for which the factors differ appreciably from unity being destroyed by mutual interference.

00Now, both the Korahite and Asaphic groups of psalms are remarkable that they hardly contain any recognition of present sin on the part of the community of Jewish faith - though they do confess the sin of Israel in the past - but are exercised with the observation that prosperity does not follow righteousness either in the case of the individual (xlix., lxxiii.) or in that of the nation, which suffers notwithstanding its loyalty to God, or even on account thereof (xliv., lxxix.).

00Malachi, Ezra and Nehemiah, like Haggai and Zechariah, are still very far from holding that the sin of Israel lies all in the past.

00Since the angles of incidence and refraction are connected by the relation sin i=µ sin r (Snell's Law), µ being the index of refraction of the medium, then the problem may be stated as follows: to determine the value of the angle i which makes D = 2 (i - r) +n (7r - 2r) a maximum or minimum, in which i and r are connected by the relation sin i =µ sin r, µ being a constant.

00They no doubt deferred the baptism which is death to sin, perhaps because, like the Cathars, they held post-baptismal sin to be unforgivable.

00The Key of Truth regards the water as a washing of the body, and sees in the rite no opus operatum, but an essentially spiritual rite in which "the king releases certain rulers a from the prison of sin, the Son calls them to himself and comforts them with great words, and the Holy Spirit of the king forthwith comes and crowns them, and dwells in them for ever."

00the moon-god Sin) it has a kind of straight brim which gives it a certain resemblance to a low-crowned " bowler."

00He held a prominent place in the New School branch of the Presbyterians, to which he adhered on the division of the denomination in 1837; he had been tried (but not convicted) for heresy in 1836, the charge being particularly against the views expressed by him in Notes on Romans (1835) of the imputation of the sin of Adam, original sin and the atonement; the bitterness stirred up by this trial contributed towards widening the breach between the conservative and the progressive elements in the church.

00His extraordinary thinness is commemorated, among other things, by the very poor but well-known epigram attributed to Young, and identifying him at once with "Satan, Death and Sin."

00of the area A, f f xdxdy = o, ffydxdy = 0, (6) R = p hA, (7) xhA = - cos a f f x 2 dA - sin affxydA, fxydA, (8) yhA = - cos a ff xydA - sin ail y 2 dA.

00(9) Turning the axes to make them coincide with the principal axes of the area A, thus making f f xydA = o, xh = - a 2 cos a, y h = - b 2 sin a, (io) where ffx2dA=Aa2, ffy 2 dA= Ab 2, (II) a and b denoting the semi-axes of the momental ellipse of the area.

00The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF', at a distance FK from F FK= (k2-hV/A)/FQ sin QFF' (2) through an angle 0 or a slope of one in m, given by P sin B= m wA FK - W'Ak 2V hV FQ sin QFF', (3) where k denotes the radius of gyration about FF' of the water-line area.

00Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = o, = o; and in steady motion the equations reduce to dH/dv=2q-2wn = 2gco sin e, (4) where B is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dv is drawn perpendicular.

00For example, a pair of equal opposite vortices, moving on a line parallel to a plane boundary, will have a corresponding pair of images, forming a rectangle of vortices, and the path of a vortex will be the Cotes' spiral r sin 20 = 2a, or x-2+y-2=a-2; (io) this is therefore the path of a single vortex in a right-angled corner; and generally.

00if the angle of the corner is jr/n, the path is the Cotes' spiral r sin n0=na.

00(22) Conjugate functions can be employed also for the motion of liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the meridian and parallel in colatitude 0 and longitude A can be written d¢_ i _ d4, I dip _ dy (13) d8 sin - 0 dX' sin 0 dX de' and then = F (tan O.

00Consider the motion given by w=U(z+a2/z), (I) 4,=U(r+- r) cos 0= U + a1x, so that (2) = U (r-)sin 0= U(i -¢) y.

00Then 4, =o over the cylinder r = a, which may be considered a fixed post; and a stream line past it along which 4, = Uc, a constant, is the curve (r - ¢2) sin 0=c, (x2 + y2) (y - c) - a 2 y = o, (3) a cubic curve (C3).

00If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = - Uz, we are left with w = Ua2/z, (6) =U(a 2 /r) cos 0= Ua2x/(x2+y2), (7) 4, = -U(a 2 /r) sin 0= -Ua2y/( x2+y2), (8) giving the motion due to the passage of the cylinder r=a with velocity U through the origin 0 in the direction Ox.

00Along the path of a particle, defined by the of (3), _ c) sine 2e, - x 2 + y2 = y a 2 ' (Io) sin B' de' _ 2y-c dy 2 ds ds' on the radius of curvature is 4a 2 /(ylc), which shows that the curve is an Elastica or Lintearia.

00When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?

00Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.

00Consider the streaming motion given by w =m =a+si, (5) 4=m ch (n -a)cos(-0), p=m sh(n-a)sin(-13).

00Over any ellipse n, moving with components U and V of velocity, =i+Uy-Vx=[msh(n-a) cos (3+Ucshn] sin k -[msh(n-a) sin (3+Vcchn] cos h; (7) so that ' =o, if U c sh n cos R, V = c ch n sin a, (8) m sh(n - a) m sh(n - a).

00An ellipse interior to n = a will move in a direction opposite to the exterior current; and when n = o, U = oo, but V = (m/c) sh a sin 13.

00The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex.

00Similarly, the streaming velocity V reversed will give rise to a thrust 27rpmV in the direction xC. Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respect +m /a) 2 - U2 The components of the liquid velocity q, in the direction of the normal of the ellipse n and hyperbola t, are -mJi sh(n--a)cos(r-a),mJ2 ch(n-a) sin (E-a).

00Next consider the motion given by = m ch 2(77a)sin 2E, tii= -m sh 2(na)cos 2E; (I) in which > ' =o over the ellipse a, and =1'+IR(x2+y2) =[ -m sh 2(7 7 -a)+4Rc 2 ]cos 4Rc2 ch 2n, (2) which is constant over the ellipse n if 4Rc 2.

00The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the ellipse n to infinity.

00x+yi =c1,1 [sin(+ 7 ni)] (17) i ' =Qc sh((n-a)sin((E-,6) (18) 'will give motion streaming past the fixed cylinder n = a, and dividing along t =43; and then x 2 -3/ 2 = c 2 sin ch n, 2xy = c 2 cos sh n.

00The polar equation of the cross-section being rI cos 19 =al, or r + x = 2a, (3) the conditions are satisfied by = Ur sin g -2Uairi sin IB = 2Uri sin 10(14 cos 18a'), (4) 1J/ =2Uairi sin IO = -U1/ [2a(r-x)], (5) w =-2Uaiz1, (6) and the resistance of the liquid is 2lrpaV2/2g.

00Similarly, with the function (19) (2n+ I) 3 ch (2n+ I) ITrb/a' (2) Changing to polar coordinates, x =r cos 0, y = r sin 0, the equation (2) becomes, with cos 0 =µ, r'd + (I -µ 2)-d µ = 2 ?-r3 sin 0, (8) of which a solution, when = o, is = (Ar'+) _(Ari_1+) y2,, ?

00For instance, with n = I in equation (9), the relative stream function is obtained for a sphere of radius a, by making it, y' =1y+2Uy 2 = 2U(r 2 -a 3 /r) sin?

000, 1, = -ZUa 3 sin e B/r; (13) and then =Ux(I+1a3/r2), 4=ZUa 3 cos 0/r2, -d r = Ua3 cos B, -d9=ZUa3 sin 0, so that, if the direction of motion makes an angle >G with Ox, tan (4y-0) =Z tan 0, tan =3 tan 0/(2-tan 2 e).

00(16) Along the path of a liquid particle 4)' is constant, and putting it equal to 2Uc2, (r 2 - a 3 /r) sin 2 0 = c 2, sin 2 0 = c2r/(r3 - a3), (17) the polar equation; or y 2 = c2r3/(r3 - a 3), r3 = a3y2 /(y2._ c2), (18) a curve of the 10th degree (C10).

00In the absolute path in space cos Ili = (2 - 3 sin 2 6)/1/ (4-sin 2 6), and sin 3 B = (y 3 -c 2 y)/a 3, (19) which leads to no simple relation.

00The velocity past the surface of the sphere is dC r sin 0 dy 2U (2r+ a 2) r sin g z U sin e, when r =a; (20) so that the loss of head is (!

00sin e 0 - i) U 2 /2g, having a maximum a U 2 /2g, (21) which must be less than the head at infinite distance to avoid cavitation at the surface of the sphere.

00Put S2 1 =12 cos 4, 12 2 = -12 sin 4, d4 d52 1 dS22 Y a2+c2 122 7Ti = 71 22 CL2- c2(121+5221)J, a2 +c2 do a2+c2 + 4c2 z dt a'-c2 (a2+,c2)2 M+2c2(a2-c2 N-{-a2+c2 2 Ý_a 2 +c 2 (' 4c2 .?"d za 2 -c 2 c2)2 2'J Z M+ -c2) which, as Z is a quadratic function of i 2, are non-elliptic so also for; G, where =co cos, G, 7 7 = - sin 4.

00Uniplanar motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the coordinates x, y by w=f(z), where z= x +yi, w=4-Plg, and then dw dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q dw u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a vector representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q.

00If there are more B corners than one, either on xA or x'A', the expression for i is the product of corresponding factors, such as in (5) Restricting the attention to a single corner B, a = n(cos no +i sin 110) _ (b-a'.0-a) +1!

00(b-a.0-a) (6) (Q) AI (a-a .0b) ch n2= ch log (Q) n cos 114+i sh log (9) re sin n9 = 2(r+ fi n) = b - a ' ju -a (7) a-a' l u-b nf2 = sh log (cos nO +1 ch log (" sin 110 =2(?"

002n u (a -b.b -a')' not requiring the integration of (II) and (12) If 0=a across the end J J' of the jet, where u = oo, q= Q, b-a' a-b ch 7/2,=cos na = I, , sh 162 = i sin na = i,, a - a a-a Then a-a'+(a+a) cos 2na-[a+a'+(a-a) cos 2najcos 2110 (a-a') sin' 2na X cos 211a - cos 2110 Along the wall AB, cos nO =o, sin n0= I, a> u> b, ch n2= i sh log (q) n= i?

00m _ c Q du dO dt - rrqu 2r qu AB _ Q du L bq (u -a') +V (b -a')1,l (a -u)11/ndu) L 1,1 (a-a/),,1 (u-b), f u Along the wall Bx, cos n0 =I, sin n0 =o, b >u>o ch nSt = ch log () n=, , fb-a?, ?

005, I ch S2 = u a, sh C2= ' u (I) (I and along the jet APJ, oo > u=aerslc>a, sh S2=i sin 0 =iu=ie zrs/o, (2) PM sin 0 ds = f e ds = 1 = 1 sin 0, (3) cos 272a - cos 2n0 = 2Q - ?ib L a b2 s i n' 27ta u-b A (a- (u -a.u -b') sin 2110 - 2 a-a .u-b ?l (u -a.u -a') = s in 2na u-b 2n b) A (ab.ba') p l u -bJ (u -a.u -a') sh nS2=i sin 110=i then the radius sin 2170 (30) A', cos nO= i, sin n0=o, n 1 ' b-a' ch nS2= ch log (9) = Va -a' n shnS2= shlog (Q) q _ o> u>a'.

00Along the jet surface A'J', q = Q, b-a' ch nSl= cos 110= a-a la - b sh nft=i sin nO=i a'>u=a'erl"> -oo, giving the intrinsic equation.

00(I) Over the jet surface 4'=m, q=Q, u=-e rr,lm= -berslc, ch SZ=cos n0= e>rsle+I, shS2 =i sin ins =tan ds 2n (3) e2 =tan nO, - c dB sin 2,10' For a jet impinging normally on an infinite plane, as in fig.

00b a-b (7) cos = a, sin na = 1j a, nd along the free surface APJ, q =Q, 4) =o, u =e-.

00u -b' Along a jet surface, q=Q, and ch S2= cos 0 =cos a-i sin2a(a-a')/(u-b), (5) if 0 =-a at the source x of the jet xB, where u = co; and supposing 0=0,13 at the end of the streams where u =j, j', u-b i sin 2 a u - j cos 0-cos /3 i a -a cos a sin a -cos 0' aa' - 2 (cos a -cos (3) (cos a-cos 0)' u-j' 1 2 cos 0-cos, (6) a -a' - 2 S i n a (cos a -cos (3') (cos a -cos B)' and 4' being constant along a stream line d4 - dw ds _d8 d4 _ dw du du du' d- -dud0' 7rQ ds_ it ds (cos a-cos /3) (cos a -cos (3') sin 0 m+m' dB c d0 - (cos a-cos B) (cos 0-cos /3) (cos 0 -cos /3')' _ sin 0 cos a-cos 13 sin 0 - cos a-cos B + cos 0-cos (3' cos 0-cos 13 cos a -cos $ sin 6 cos (3-cos /3' cos 0-cos 0" giving the intrinsic equation of the surface of a jet, with proper attention to the sign.

00From A to B, a>u >b, 0=0, ch S2= ch log Q=cos a-i sin 2a a-b I sh S2= sh log Q= I (a u-b-a/) s i n a Q = (u-b) cos a-2(a-a') sin 2 a+1,/ (a-u.u- a')sin a (8) u-b ds _ ds d4 _ Q dw Q du - Q d 4) du q du (u-b) cos a-2(a- a') sin 2 a (a-u.0 - a') sin a (9) it j- -j' AB _f a(2b - a - a')(u-b)-2(a-b)(b-a')+2V (a - b.

008 (2) (4) e, rs /c e ns/c + I' (2) cos n0= cos na-N e' 31 ' - cos'na' cos 2 na sin2n0 (8) sin 2 n0 - sin2na' he intrinsic equation, the other free surface A'P'J' being given by e m /?

00- cos 2 na sin2ng s (9) sin 2 na - sin2n0 Putting n =I gives the case of a stream of finite breadth disturbed y a transverse plane, a particular case of Fig.

00Trans., 1890); the solution is given by ch nS2=sn w, shnS2=i cn w (II) so that, round the boundary of the polygon, ik = K', sin n8 =o; and on the surface of the vortex 1P= o, q = Q, and cos n8=sn4p,nB= Zit -am sic, (12) the intrinsic equation of the curve.

00Introducing Euler's angles 0, c15, x1= F sin 0 sin 0, x 2 =F sin 0 cos 0, xl+x 2 i =iF sin 0e_, x 3 = F cos 0; sin o t=P sin 4+Q cos 0, dT F sin 2 0d l - dy l + dy 2x = (qx1+ryi)xl +(qx2+ry2)x2 = q (x1 2 +x2 2) +r (xiyi +x2y2) = qF 2 sin 2 0-Fr (FG - x 3 y 3), (16) _Ft (FG _x 323 Frdx3 (17) F x3 X3 elliptic integrals of the third kind.

00well as of the body from the vector OF to O'F' requires an impulse couple, tending to increase the angle F00', of magnitude, in sec. foot-pounds F.00'.sin FOO'=FVt sin (0-0), (4) equivalent to an incessant couple N=FV sin (0-0) = (F sin 0 cos 0-F cos 0 sin ¢)V = (c 2 -c i) (V /g) sin 0 cos 4) =W'(13-a)uv/g (5) This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c 2 -c 1 is positive to set the body broadside to the advance.

00Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented by (1) c 1 =W+W'a, c2=W+W'/3' where a, R are numerical factors depending on the external shape; and if the C.G is moving with velocity V at an angle 4) with the axis, so that the axial and broadside component of velocity is u = V cos 0, v =V sin 4), the total momentum F of the medium, represented by the vector OF at an angle 0 with the axis, will have components, expressed in sec. Ib, F cos 0 =c 1 - = (W +W'a) V cos 43, F sin 0 = c 2.11 = (W +W'/3) V sin 4) .

00For the body alone the resultant of the components of momentum W V -cos andW V sin 0 is W V -sec. lb, acting along 00', and so is unaltered.

00the moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o.

00As the ring is moved from 0 to 0' in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from aM'U +E and SM'V along Ox and Oy to aM'U'+ and /3M'V' along O'x' and O'y', (I) the axis of the ring changing from Ox to O'x'; and U = Q cos 0, V = Q sin 0, U' =Q cos (o - Rt), V' =Q sin (0 - Rt), (2) so that the increase of the components of momentum, X 1, Y 1, and N1, linear and angular, are X 1 = (aM'U'+ 0 cos Rt - aM'U - - 1 3M'V' sin Rt =(a - (3)M'Q sin_(0 - Rt) sin Rt - ver Rt (3) Y 1 = (aM'U'+) sin Rt-[-13M'V' cos Rt - (3M'V = (a - (3) M'Q cos (0 - Rt) sin Rt +t sin RT, N1=[ - (aM'U'+E) sin (0 - Rt)+ 1 3M'V' cos (o - Rt)]OO' = [- (a - 1 3) M'Q cos (o - Rt) sin (o - Rt) - sin (o - Rt) ]Qt.

00X i /t=(a - (3)M'QR sin 0= (a - (3)M'VR, (6) Y=It.

00ZI /t = - (a - s) M'Q 2 sine cos ° - EQ sin() =[ - (a - (3)M'U+E]V (8) Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid - aM' (Ç_vR), - (3M' (-- E -FUR), - EC' dR, (9) so that its equations of motion are M (Ç - vR) _ - aM' (_vR) - (a - $) M'VR, (io) M (Ç+uR) = - OM' (dV+U R) - (a - ()M'UR - R, '(II) C dR = dR + (a - Q)M'UV+0V; (12) and putting as before M+aM'=ci, M+13M' = c2, C+EC'=C3, ci dU - c2VR=o, dV +(c1U+E)R=o, c 3 dR - (c 1 U+ - c 2 U)V =o; showing the modification of the equations of plane motion, due to the component E of the circulation.

00The integral of (14) and (15) may be written ciU+E=Fcoso, c 2 V= - Fsino, dx F cost o F sinz o 71 = U cos o - V sin o = cl + c c ic os o, chi = U sine +V coso= (F - F) sin cos o - l sino, (19) c i 2 2 2 sin o cos o - l ?

00sin o= F dl, (20) C3 do F2 h _ F2 cos 2 o F 2 sin z o F dt y - V C G c +2 c1 coso+H]; (21) 1 z so that cos 0 and y is an elliptic function of the time.

00Another son was high-priest of the city of Tutu, and in the name of his daughter, Lipus-Eaum, a priestess of Sin some ur dynasty.

00Copper Vot1 Sin, King Of Larsa.

00The last king of Assyria was probably the brother of Assur-etil-ilani, Sin - sar - iskun (Sin-sarra-uzur), who seems to have been the Sarakos (Saracus) of Berossus.

00The moon-god Sin is written by a sign which has the force of " thirty," and is a distinct reference to the monthly course of the planet; or the name is written by two signs to be pronounced EN-ZU, which describe the god as the " lord of wisdom."

001) and treat of the sin of the angels that led to the flood, and of their temporal and eternal punishment.

00The book treats of the Messiah and the Messianic kingdom, the woes of Israel in the past and the destruction of Jerusalem in the present, as well as of theological questions relating to original sin, free will, works, &c. The views expressed on several of these subjects are often conflicting.

00It is possible for a regenerate man to live without sin.

00brings the prophet a new sense of sin as essentially a matter of the heart, and an awakened conscience as before the " glory of God," the Creator and Upholder of all things.

00But, while Samaria is summarily dismissed, the sin of Judah is analysed at length in chs.

00At last the power and influence of the spirits of darkness, with whom man associates himself by his sin, became so great that the existence of the human race was threatened, and Jehovah was necessitated to descend into nature to restore the connexion between Himself and man.

00The romantic school was supported by Sin Otetchestva (1812), " Son of the Fatherland," united in 1825 to the Severnoi Arkhiv (1822), which dwindled and came to an end soon after 1839.

00451 affirmed "that Christ is true God and true man, according to His Godhead begotten from eternity and like the Father in everything, only without sin; and that after His incarnation the unity of the person consists in two natures which are conjoined without confusion, and without change, but also without rending and without separation."

00In elliptic integrals, the amplitude is the limit of integration when the integral is expressed in the form f 4) 1% I - N 2 sin e 4) d4.

00Its preaching is practical and direct, asseverating the reality of Sin, "the everlasting punishment of the wicked," and Redemption.

00It ex pounded in terse and significant teaching the doctrine (1) of God, (2) of original sin, (3) of the Son of God, (4) of justification..

00It deals with the Bible as the final appeal in controversy, the doctrines of God, man, sin, the Incarnation, the Resurrection of our Lord Jesus Christ, " both the Son of man and the Son of God," the work of the Holy Spirit, justification by faith, the perpetual obligation of Baptism and the Lord's Supper, final judgment, the law of Christian fellowship. The same principles have been lucidly stated in the Evangelical Free Church catechism.

00Sin; 9.

00He knows that the great object of all my preaching and writing was to convert men from sin.

00The church key reminds him that "it is my sin that locks his handes," and the stones of the floor are patience and humility, while the cement that binds them together is love and.

00Of in Jesus Christ the Saviour, who delivers from the bondage of sin by his life, doctrine and death; in the operation of the Holy Ghost; in a holy, universal, Christian church; in forgiveness of sins and the life everlasting.

00The latter enunciated the following rule: " If a bishop or priest be living in mortal sin, then he neither ordains, nor consecrates, nor baptizes."

00The Cathars even held it necessary, in case a bishop fell into mortal sin, to repeat his ba p tisms and ordinations, for they had been vitiated by his sins.

00Although they surrendered transubstantiation, the loss of one mystery was amply compensated by the stupendous doctrines of original sin, redemption, faith, grace and predestination upon which they founded their theory of salvation.

00If he be once approved as a true prophet, his words and acts are not to be criticized; for this is the sin that shall not be forgiven.

00The Lombard sect went farther in (3) and (4), holding that no one in mortal sin could consecrate the sacrament, and that the Roman Church was the scarlet woman of the Apocalypse, whose precepts ought not to be obeyed, especially those appointing fast-days.

00Let a be the radius of a circle, and 0 (circular measure) the unknown angle subtended by an arc. Then, if we divide 0 into m equal parts, and L 1 denotes the sum of the corresponding chords, so that L i =2ma sin (0/2m), the true length of the arc is L1 +a9 3 - 5 + ..., where cp. =B/2m.

00Jesus, they say, was sent to save sinners; was he not sent to help those who have kept themselves free from sin?

0046) and how blood and water flow from His spear-pierced side: thus the Lamb " taketh away the sins of the world " by shedding His blood which " cleanseth us from every sin "; and " He cometh by water and blood," historically at His baptism and crucifixion, and mystically to each faithful soul in baptism and the eucharist.

007, 8); " they have no excuse for their sin " (xv.

00There are twenty-nine letters, one more than in Arabic, Samech and Sin being distinct forms, as in Hebrew.

00The moon-god Sin appears on an inscription of Shahwat; but, according to Hamdani, Haubas, " the drier," was the Sabaean moon-god.

00On the Shabwat inscription `Athtar is the father of Sin, and it is noteworthy that these two deities also appear as nearly related in the Babylonian legend of 'Ishtar's descent to Hades, where `Ishtar is conversely the daughter of the god Sin.

00y=a sin (x - Ut).

00The simplest form of wave, so far as our sensation goes - that is, the one giving rise to a pure tone - is, we have every reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y=a sin m(x - e).

00If we put this in the form y=a sin (x - e), we see that y=o, for x =e, e +aX, e+2A, e+;X, and so on, that y is + from x=e to x=e+iX, - from e+zX to e+?A, and so on, and that it alternates between the values+a and - a.

00If we transfer 0 to A, e=o, and the curve may be represented by y=a sin A x.

00If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e =OA at any time t will be Ut, so that the value of y may be represented as 2 y=a sin T (x - Ut).

00We may therefore put y=a sin -T (x - e) +b sin (x - f)+c sin 6 (x - g) +&c. (21) where the terms may be infinite in number, but always have wavelengths submultiples of the original or fundamental wave-length A.

00Putting A /M =n 2 the equation becomes x+n 2 x=o, whence x =A sin nt, and the period is 27r/n.

00Repre - senting it by -P sin pt, the equation of motion is now 2 -M sin pt=o.

00(22) Let us assume that the body makes vibrations in the new period 27rp, and let us put x = B sin pt; substituting in (22) we have p 2 B +n 2 B +P/M =0, whence P r B - M p2 _ n2 and the " forced " oscillation due to -P sin pt is x = P

00We may represent the displacement due to one of the trains by y l =a sin 2 i (24) where x is measured as in equation (16) from an ascending node as A in fig.

00If we measure t from an instant at which the two trains exactly coincide, then as U for the other train has the opposite sign, its displacement is represented by y2= a sin (x+Ut).

00The velocity of the string at P parallel to PM is U sin 4), and the mass of string passing P is mU per second, so that mU 2 sin is carried out per second.

00But the tension at P is T, parallel to the tangent, and T sin 4 parallel to PM, and through this - T sin is the momentum passing out at P per second.

00Since the resultant is zero, mU 2 sin 4) - T sin ct.

00If then y=a sin (x - Ut),, t=2,ro-ra cos (x - Ut).

00If we measure the time from an instant at which the two are in the same phase the resultant disturbance is y=a sin i t+a sin 27rn2t =2a cos ir(n i - n 2)t sin ir(nl-t-n2)t, which may be regarded as a harmonic disturbance of frequency (ni+n2)/2 but with amplitude 2a cos 7r(n i - n 2)t slowly varying with the time.

00Let us suppose that with constant excess of pressure, p, in the wind-chest, the amplitude produced is proportional to the pressure, so that the two tones issuing may be represented by pa sin 27-nit and pb sin 21rn 2 t.

00Then the issuing disturbance will be }p+Aa sin (27rn i t+e)+µb sin (271-n 2 t+f)1 {a sin 22rn,t+b sin 27rn20 =pa sin 21rn i t+pb sin 21rn2t (1.

00Suppose now that F =a sin 22rn i t+b sin 21rn 2 t, the second term will evidently produce a series of combination tones of periodicities 2n 1, 2772, n, - n2, and n 1 -1-n 2, as in the first method.

00But certainly the myth does help us to imagine a story in which, for some sin against the gods, some favoured hero was hurled down from the divine abode, and such a story may some day be discovered.

00It is held that, but for his sin, Adam would have been immortal.

004) the sin of Adam was the cause of physical death; according to another (liv.

00Truly, a worthy development of the seed-thoughts of the original narrator, and (must we not add ?) entirely opposed to any doctrine of Original Sin.

00One is that Adam is said to have had from the first a wicked heart, owing to which he fell, and his posterity likewise, into sin and guilt.

00All men have the same seed of evil in them that Adam had; they sin and die, like him.

00The first ended with the Flood, so that any consequences of Adam's sin were, strictly speaking, of limited duration.

0022, 45, 47), whose doctrine of sin is evidently of mixed origin.

00On the belief in the Fall, see Tennant, The Sources of the Doctrine of the Fall and Original Sin (1903).

00In later times Ate is regarded as the avenger of sin (Sophocles, Antigone, 614, 625).

00It refers not to an accusation, but to sin actually committed (after baptism); and it denotes the setting of the sinner free from the guilt of the sin, or from its ecclesiastical penalty (excommunication), or from both.

00sin may be forgiven; and this restoration is the atonement.

00The Old Testament has no theory of sacrifice; in connexion with sin the sacrifice was popularly regarded as payment of penalty or compensation.

00The idea of vicarious atonement appears in the Old Testament in different forms. The nation suffers for the sin of the individual; 8 and the individual for the sin of his kinsfolk 7 or of the nation.

002, " sin unwittingly " bishegaga, c. 450 B.C., &c.

00Christ suffering on behalf of sinners satisfies the divine righteousness, which was outraged by their sin.'3 His work is an expression of God's love to man; 14 the redeeming power of Christ's death is also explained by his solidarity with humanity as the second Adam," - the redeemed sinner has " died with Christ."

0018 In John, Christ is a " propitiation " (tXaa l uos) provided by the love of God that man may be cleansed from sin; He is also their advocate (HapetOsi ros) with God that they may be forgiven, for His name's sake.

002 The leading reformers emphasized the idea that Christ bore the punishment of sin, sufferings equivalent to the punishments deserved by men, a view maintained later on by Jonathan Edwards junior.

00For instance, it has been held that Christ atoned for man k ind not by enduring the penalty of sin, but by identify views.

00Y Y ing Himself with the sinner in perfect sympathy, and feeling for him an " equivalent repentance " for his sin.

00(and) because in His submission to the awful penalty of sin.

00there was a revelation of the righteousness of God, which must otherwise have been revealed in the infliction of the penalties of sin on the human race."

00towards God because it expressed His perfect obedience, it manifested God's righteous wrath against sin, and in virtue of Christ's human nature involved man's recognition of the righteousness of God's condemnation of sin; also because in some mysterious way death has a propitiatory value; and finally because Christ is the representative of the human race.

00At present the belief in an objective atonement is still widely held; whether in the form of penal theories - the old forensic view that the death of Christ atones by paying the penalty of man's sin - or in the form of governmental theories; that the Passion fulfilled a necessity of divine government by expressing and vindicating God's righteousness.

00After the fall of Ayuthia a great general, Phaya Takh Sin, collected the remains of the army and restored the fortunes of the kingdom, establishing his capital at Bangkok; but, becoming insane, he was put to death, and was succeeded by another successful general, Phaya Chakkri, who founded the present dynasty.

00For the simplest case of polarized waves travelling parallel to the axis of x, with the magnetic oscillation y along z and the electric oscillation Q along y, all the quantities are functions of x and t alone; the total current is along y and given with respect to our moving axes by __ (d_ d Q+vy d K-1 Q, dt dx) 47rc 2 + dt (4?rc 2) ' also the circuital relations here reduce to _ dydQ _dy _ dx 47rv ' _ dt ' d 2 Q dv dx 2 -417t giving, on substitution for v, d 2 Q d 2 Q d2Q (c2-v2)(7372 = K dt 2 2u dxdt ' For a simple wave-train, Q varies as sin m(x-Vt), leading on substitution to the velocity of propagation V relative to the moving material, by means of the equation KV 2 + 2 uV = c 2 v2; this gives, to the first order of v/c, V = c/K i - v/K, which is in accordance with Fresnel's law.

00Meanwhile in the Western Church the subject of sin and grace, and the relation of divine and human activity in salvation, received especial attention; and finally, at the second council of Orange in 529, after both Pelagianism and semi-Pelagianism had been repudiated, a moderate form of Augustinianism was adopted,, involving the theory that every man as a result of the fall is in such a condition that he can take no steps in the direction of salvation until he has been renewed by the divine grace given in baptism, and that he cannot continue in the good thus begun except by the constant assistance of that grace, which is mediated only by the Catholic Church.

00Since the landed proprietors disposed of churches and convents, and the kings of bishoprics and abbeys, it became possible for them too to commit the sin of simony; hence a final expansion, in the iith century, of the meaning of the term.

009; no doubt also money, a sin the Canaanite temple at Shechem, Judges ix.

00(3) Then a fresh controversy was started by the assertion that sin was part of the substance of man in his fallen condition.

00It was answered that sin had not totally destroyed man's ethical nature, and that grace changed what was morally insensitive into what was morally sensitive, so that there could be a cooperation between God's grace and man's will.

00indulgentia, indulgere, to grant, concede), in theology, a term defined by the official catechism of the Roman Catholic Church in England as " the remission of the temporal punishment which often remains due to sin after its guilt has been forgiven."

00For some purposes it is preferable to retain the circular measure, i radians, as being undistinguishable from sin i and tan i when i is small as in direct fire.

00is given by sin S=h/3R, (41) or, for a small angle, expressed in minutes, taking the radian as 3438', (42) S = 1146h/R.

00di g d tan i g dt - v cos i ' and now (53) dx d 2 y dy d2xdx Cif dt 2 dt dt2 _ - _ gdt' and this, in conjunction with (46) dy _ d y tan i = dx dt/dt' (47)di d 2 d d 2 x dx sec 2 idt = (ctt d t - at dt2) I (dt), reduces to (48) Integrating from any initial pseudo-velocity U, (60) du t _ C U uf(u) x= C cos n f u (u) y=C sin n ff (a); and supposing the inclination i to change from 0, to 8 radians over the arc.

00The difficulty is avoided by the use of Siacci's altitude-function A or A(u), by which y/x can be calculated without introducing sin n or tan n, but in which n occurs only in the form cos n or sec n, which varies very slowly for moderate values of n, so that n need not be calculated with any great regard for accuracy, the arithmetic mean 1(0+0) of ¢ and B being near enough for n over any arc 4)-8 of moderate extent.

00- 1 or, as (88) and (90) may be written for small angles, (91) sin 20.=2C [I (V) - oS j, (92) sin 20 =2C [O S - I (v)] To simplify the work, so as to look out the value of sin 20 without the intermediate calculation of the remaining velocity v, a doubleentry table has been devised by Captain Braccialini Scipione =S (U) - S (u), = I (U) - I (u); mean angle (70), (Problemi del Tiro, Roma, 1883), and adapted to yd., ft., in.

00In this table (93) sin 20=Ca, where a is a function tabulated for the two arguments, V the initial velocity, and R/C the reduced range in yards.

00The mastery which he had obtained over the mathematical symbols was so complete that he never shrank from the use of expressions, however complicated - nay, the more complicated they were the more he seemed to revel in them - provided they did not sin against the ruling spirit of all his work - symmetry.

00This writer, who is characterized by a singularly bright and picturesque style, and also by deep religious feeling and insight, begins his narrative with the account of the creation of man from the dust, and tells of the first sin and its consequences (Gen.

00Jeremiah was keenly conscious of his people's sin; and the aim of most of his earlier prophecies is to bring his countrymen, if possible, to a better mind, in the hope that thereby the doom which he sees impending may be averted - an end which eventually he saw clearly to be unattainable.

00The dominant theory at the time when Job was written was that all suffering was a punishment of sin; and the aim of the book is to controvert this theory.

00and Sin.

00Again, since through Eve sin entered into the world, woman must be subordinate to man (I Tim.

0021-23, as also in contemporary Jewish thought; and by other combined quotations Paul enunciates the universality of sin (Rom.

00As a presumptive ruler of England she was, like Cecil, and for that matter the future archbishop Parker also, too shrewd to commit herself to passive or active resistance to the law; and they merely anticipated Hobbes in holding that the individual committed no sin in subordinating his conscience to the will of the state, for the responsibility for the law was not his but the state's.

00To the mass these signs are unintelligible, because they deem it impossible that Yahweh should utterly cast off His chosen nation; but to those who know His absolute righteousness, and confront it with the people's sin, the impending approach of the Assyrian can have only one meaning and can point to only one issue, viz.

00The author of the Otbaxr7 goes so far as to assert that whoever does this is guilty of the sin against the Holy Ghost.

00By means of ascetic observances the man becomes once more a spiritual and enduring being, free from all sin.

00The logarithm is also a function of frequent occurrence in analysis, being regarded as a known and recognized function like sin x or tan x; but in mathematical investigations the base generally employed is not 10, but a certain quantity usually denoted by the letter e, of value 2.71828 18284 ...

00The mathematical function log x or log x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin-' x,&c., log x and e x which are universally treated in analysis as known functions.

00It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(I +x) =x -",, x2 +3x 3 -4x 4 + is true only when the analytical modulus of x is less than unity.

00Also exp (E+in) =e (cos 7 7 +i sin 7,).

002 The method consists in the use of the formula sin a sin b=2 {cos(a-b)-cos(a+b)l, by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only.

00Under these heads it discusses respectively the sin and misery of men, the redemption wrought by Christ (here are included the Creed and the Sacraments), and the grateful service of the new life (the Decalogue).

00The last book of the Laws of Manu deals with karmaplialam, " the fruit of karma," and gives many curious details of the way in which sin is punished and merit rewarded.

00Ignatius distinctly excepts the case where obedience in itself would be sinful: "In all things except sin I ought to do the will of my superior and not my own."

00From this it is clear that only in doubtful cases concerning sin should an inferior try to submit his judgment to that of his superior,to be not only one who would not order what is clearly sinful, but also a competent judge who knows and unds, better than the inferior, the nature and aspect of the command.

00Alexander condemned in 1690 the doctrines of so-called philosophic sin, taught in the Jesuit schools.

00These were: (1) that the divine decree of predestination is conditional, not absolute; (2) that the Atonement is in intention 'universal; (3) that man cannot of himself exercise a saving faith; (4) that though the grace of God is a necessary condition of human effort it does not act irresistibly in man; (5) that believers are able to resist sin but are not beyond the possibility of falling from grace.

00The bread and wine are indeed an offering to God of what is his own, pure because offered in purity of heart; but they are not interpreted of the sacrifice of Jesus' body broken on the cross, or of his blood shed for the remission of sin.

00And this food is called by us Eucharistia, and of it none may partake save those who believe our teachings to be true and have been washed in the bath which is for remission of sin and rebirth, and who so live as We should probably omit the words bracketed.

00In these centuries baptism was the rite for the remission of sin, not the Eucharist; it is the prophet in the Didache who presides at the Lord's Supper, not the Levitically conceived priest; nor as yet has the Table become an Altar.

00in the section on the Mass merely protests against the view that " the Lord's Supper is a work (opus) which being performed by a priest earns remission of sin for the doer and for others, and that in virtue of the work done (ex opere operato), without a good motive on the part of the user.

00Christ who knew no sin (2 Cor.

0021) had been made sin, and sacrificed for us, becoming as it were a new Passover (1 Cor.

00This reconstruction of its meaning seems to have been the peculiar revelation of the Lord to Paul, who viewed Christ's crucifixion and death as an atoning sacrifice, liberating by its grace mankind from bonds of sin which the law, far from snapping, only made more sensible and grievous.

00The first man, Adam, was engendered by Satan in conjunction with "sin," "cupidity," "desire."

00Instead of the subtle Catholic theories concerning divine predestination and human freedom, and instead of a difficult theodicaea, it offered an exceedingly simple conception of sin and goodness.

00The self-conjugate circle is a 2 sin 2A +0 2 sin 2 B +y 2 sin 2C = o, or the equivalent form a cosAa 2 +bcosB(2 +ccosCy 2 = o, the centre being sec A, sec B, sec C.

00- N / (z cot IC) =o, with centre sin A, sin B, sin C; the escribed circle opposite the angle A is - N I (- x cot ZA)+ -1 (y tan 2B) + -V (z tan 2C) =o, with centre - sin A, sin B, sin C; and the selfconjugate circle is x 2 cot A+y 2 cot B+z 2 cot C =o, with centre tan A, tan B, tan C. Since in areal co-ordinates the line infinity is represented by the equation x+y+z=o it is seen that every circle is of the form a 2 yz+b 2 zx+c 2 xy+(lx+my+nz)(x+y+z) = o.

00Comparing this equation with ux 2 +vy 2 +w2 2 +22G'y2+2v'zx+2W'xy=0, we obtain as the condition for the general equation of the second degree to represent a circle :- (v+w-2u')Ia 2 = (w +u -2v')/b2 = (u+v-2w')lc2 In tangential q, r) co-ordinates the inscribed circle has for its equation(s - a)qr+ (s - b)rp+ (s - c) pq = o, s being equal to 1(a +b +c); an alternative form is qr cot zA+rp cot ZB +pq cot2C =o; Tangential the centre is ap+bq+cr = o, or sinA +q sin B+rsinC =o.

00Exact formulae are: - Arc =a0, where 8 may be given directly, or indirectly by the relation c=2a sin 28.

00Exact formulae are: - Area = 2a 2 (0 - sin 0)=1a 2 0 - 4c2 cot zB =Za 2 -2 c?1 (a 2 -4c 2).

00The theorem for angle-bisection which Vieta used was not that of Archimedes, but that which would now appear in the form I - cos 0 = 2 sin e 20.

00II) whose centre is 0, AC its chord, and HK the tangent drawn at the middle point of the arc and bounded by OA, OC produced, then, according to Archimedes, AMC AC. In modern trigonometrical notation the propositions to be compared stand as follows: 2 tan 20 >2 sin 28 (Archimedes); tan 10+2 sin 3B>0> 3 sin B (Snell).

00As a thinker he was not original, and even as a theologian he has produced but few schemes of doctrine, except his doctrine of sin.

00A homogeneous oscillation is one which for all time is described by a circular function such as sin(nt+ a), t being the time and n and a constants.

00Glancing over its pages, however, it seemed to him a sin that a book so holy - and so saleable - should be destroyed.

00Schelling perceived that Hegel, in reducing everything to infinite mind, absorbed man's free but finite personality in God, and, in declaring that everything real is rational, failed to explain evil and sin: indeed, the English reader of T.

00Omitting correction terms depending on the temperature and on the inductive effect of the earth's magnetism on the moment of the deflecting magnet, if 0 is the angle which the axis of the deflected magnet makes with the meridian when the centre of the deflecting magnet is at a distance r, then zM sin B=I+P+y2 &c., in which P and Q are constants depending on the dimensions and magnetic states of the two magnets.

00Their special danger, then, the sin which deceived(iii.

00In its most simple and attractive form - one at the same time invested with the authority of the reputed holy author - their account of the creation of the world and of man; the origin of sin and redemption, the history of the Cross, and the disputes between body and soul, right and wrong, heaven and hell, were embodied either in "Historiated Bibles" (Paleya 1) or in special dialogues held between Christ and his disciples, or between renowned Fathers of the Church who expounded these views in a simple manner adapted to the understanding of the people (Lucidaria).

00the god Sin), and identifies it with the'AlAulPcvos of Berossus.

00The importance of IIarran was doubtless due not only to its fame as a seat of the Moon-god Sin, honoured also west of the Euphrates, and to its political position, but also to its trade relations.

00Before long, however, the overthrow of Astyages by Cyrus cleared Mesopotamia, and Nabonidus (Nabu-naid) was able, drawing on the resources of the whole of Syria for the purpose, to restore the famous temple of Sin at Harran, where a few years later he erected in memory of his mother, who seems to have been a priestess there, the stele published in 1907 by Pognon.

00Birney and His Times (New York, 1890), by his son, William Birney; and his principal writings: On the Sin of Holding Slaves (1834).

00It was a medicine of sin that could only be used once and not a second time.

00Soon afterwards he went to Saumur, where in 1679 were published Liberii de Sancto Amore Epistolae Theologicae (Irenopoli: Typis Philalethianis), usually attributed to him; they deal with the doctrine of the Trinity, the hypostatic union of the two natures in Jesus Christ, original sin, and the like, in a manner sufficiently far removed from that of the conventional orthodoxy of the period.

0017, now `ArVa), Sin (Assyr.

00Occasionally we know what the name was; the Baal of Tyre was Melqarth (Melkarth), which again means merely " king of the city "; similarly among the Aramaeans the Ba'al of Harran was the moon-god Sin.

00His writings are described by Harnack as a curious mixture of Catholic orthodoxy and unconscious tendencies to Protestantism; their most noticeable point is the great importance they attach to the fact of sin, both original and actual.

00Following Parmenides, Philolaus regarded the soul as a "mixture and harmony" of the bodily parts; he also assumed a substantial soul, whose existence in the body is an exile on account of sin.

00The prophets had an ethical conception of Yahweh; the sin of His own people and of other nations called for His intervention in judgment as the moral ruler of the world.

0017) and the Man of Sin (2 Thess.

00"After God has changed eternal punishments into temporary, the justified must expiate these temporary penalties for sin in purgatory" (p. 268).

00(5) An inadequate sense of sin.

00Julian of Indiana, were nominated for the presidency and the vice-presidency respectively, on a platform which declared slavery "a sin against God and a crime against man," denounced the Compromise Measures of 1850,1850, the fugitive slave law in particular, and again opposed the extension of slavery in the Territories.

00On some other points, as the nature of matter, the immortality of the soul and the principle of sin, his views are interesting.

00"It will be no light sin for us if we thrust out of the oversight i those who have offered the gifts unblameably and holily.

00Mysticism may broadly be described as the effort to give effect to the craving for a union of the soul with the Deity already in this life; and asceticism as the effort to give effect to the hankering after an ever-progressive purification of the soul and an atoning for sin by renunciation and self-denial in things lawful.

00I); "He hath not given any man licence to sin" (xv.

00He is a pessimist as regards women; "From a woman was the beginning of sin; and because of her we all die" (xxv.

00Merchants are expected to cheat; "Sin will thrust itself in between buying and selling" (xxvii.

00To illustrate the main features of the calculation, we may suppose that the surface is subject to a simple-harmonic cycle of temperature variation, so that the temperature at any time t is given by an equation of the form 0 - 0 0 = Asin 27rnt= A sin 27rt/T, (5) where 0 0 is the mean temperature of the surface, A the amplitude of the cycle, n the frequency, and T the period.

00The wave at a depth x is represented analytically by the equation 0 - 0 0 = Ae mx sin (21rnt - mx).

00We thus obtain the differential equation gk(d 2 0/dx 2) =cgdo/dt+hpo, which is satisfied by terms of the type =c" sin where a 2 -b 2 = hp/qk, and ab = urnc/k.

00Such an inference is, however, clearly at variance with the whole doctrine of sin, repentance and the atonement, as also with that of eternal reward and punishment, which postulates a real measure of human responsibility.

00For in fixity and resoluteness of character he likened himself to God; and having kept himself free from sin was united with God, and was empowered to grasp as it were the power and authority of wonders.

00His strength lay in his intense conviction of an intimate connexion between sin and punishment and in his power of dramatic presentation.

00He delivered the second series of the Croall lectures, on the Doctrine of Sin, which were afterwards published.

00The word is also used for a man's evil genius, which drives him to sin without any provocation; a man so driven is sometimes called Alastor.

00The process of human development Rothe regards as necessarily taking an abnormal form and passing through the phase of sin.

00In consequence of His supernatural birth the Saviour, or the second Adam, was free from original sin.

00(Fn1) Their tenets were as follows: The evil god, Satan, who inspired the malevolent parts of the Old Testament, is god and lord of this world, of the things that are seen and are temporal, and especially of the outward man which is decaying, of the earthen vessel, of the body of death, of the flesh which takes us captive under the law of sin and desire.

00Imprisoned in the garment of flesh, burdened with its sin, souls long to be clothed upon with the habitations they left in heaven.

00The consolamentum removes original sin, undoes the sad effects of the primal fall, clothes upon us our habitation which is from heaven, restores to us the lost tunic of immortality.

00The nature of the results will best be explained by reference to the formula given by Lovering as a mean from all the stations considered, viz.: - 33+3 3 sin (30t+100°52') +2 53 sin (60t+309°5') +o 16 sin (90t+213°31') +o 56 sin (120t+162°45') +0 27 sin (150t+32°38').

00In historic times the chisels are about I X3/4, X6 to Sin, long (34).

0