ao Sentence Examples

• ao-Kutani, so called because of a green (ao) enamel of great brilliancy and beauty which was largely used in its decoration, and Kirtani with painted and enamelled pate varying from hard porcelain to pottery.

• ao-Kutani, so called because of a green (ao) enamel of great brilliancy and beauty which was largely used in its decoration, and Kirtani with painted and enamelled pate varying from hard porcelain to pottery.

• 000 English Miles ao 20 30 an 5 c+ ?

• ao.

• Every plant is constrained to carry Out its functions of germination, growth, nutrition, reproduction, &c., between certain limits of temperature, and somewhere between the extremes of these limits each function finds ao optimum temperature at which the working of the living machinery is at its best, and, other things being equal, any great departure from this may induce pathological conditions; and many disasters are due to the failure to provide such suitable temperaturese.g.

• ao; Dan.

• p. 256 (19 Ao).

• We have to multiply a01; -alas+a2 by ao, -aif32+a2 and we obtain ao (3 - aoal(f31N2 +01133) +aoa2(SI+13) -i-a?31a2 - aIa2(31 + 02) + al, 131+02 = b, 131132 = b t'i +s2 = 2bob2, and clearing of fractions R 1,5 = (a o b 2 - a2 b o) 2 + (a i b o - aobi) (aib2 - a2b1).

• and by elimination we obtain the resultant ao 0 bo 0 0 al ao b1 bo 0 a 2 a i b 2 b 1 bo a numerical factor being disregarded.

• 1 X discriminant of f = ao X disct.

• This arises from the circumstance that the general operator Ao,a0aa1 + ialaa2 + 2a2 a 3 +...

• is transformed into the operator d 1 by the substitution (ac, al, a2, ï¿½ï¿½ï¿½as, ï¿½ï¿½ï¿½) _ (ao, Xoai, X 6 X i a 2, ï¿½ï¿½ï¿½, XcX1..%s_las,ï¿½ï¿½ï¿½), so that the theory of the general operator is coincident with that of the particular operator d1.

• For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x-h for x causes ao, a1, a 2 a3,...

• to become respectively ao, ai+hao, a2+2ha1, a 3 +3ha 2, ...

• ao -ialaan+l m !

• ao (m -2) m !

• being formed, the expansion being carried out, an operand ao or bo or co ...

• F(a ' a ' a, ...a) =r A F(ao, a1, a2,ï¿½ï¿½ï¿½an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation.

• ;51, 2) = r F(ao, al, a2,...; xi, x2), the function F(a 01 a 1, a 2, ...

• Instead of a single quantic we may have several f(ao, a1, a2...; x1, x2), 4 (b o, b1, b2,...; x1, x2), ...

• For the substitution rr xl =A 11 +1 2 12, 52=A21+ï¿½2E2, of modulus A1 ï¿½i = (Alï¿½.2-A2ï¿½1) = (AM), A 2 ï¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lï¿½l +ai(A1/.22+A2ï¿½1) +7,2X2/22, A2 = aoï¿½l +2a1ï¿½1/ï¿½2 +a 2ï¿½2 ï¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?

• A 1, A2 ï¿½ Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (alï¿½l +a2ï¿½2) = aAaï¿½, A 2 = (alï¿½l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a ï¿½ = a1/ï¿½1 + a21=2 respectively.

• By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1ï¿½1 +a2m2) = aa a ï¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alï¿½1+a2ï¿½2) = a A ï¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a ï¿½ respectively.

• For, if c(ao i ...x l, x 2) be a covariant of order e appertaining to a quantic of order n, t (T.

• 1 2) = (A /-?) ' (ao,...

• The Binary Quartic.-The fundamental system consists of five forms ax=f; (f,f')2=(ab) 2axbx=Ax; (f,f')4=(ab) 4= 2; (f, 0)1= (ao) azsi = (ab) 2 (cb) a:b x c5 =1; (f 4) 4 = (as) 4 = (6) 2 00 2 (ca) 2 = j, viz.

• l aa k -x 2 d d- = 0; Z(nk)ak+l adk - x ldd2=0; or in the form d d 52-x 2(7 =0, O - x1ax2 = 0; where 0 = ao d a l + 2a 1 -?...+na,,_id an, 0 = nal dao -?

• One advantage we have obtained is that, if we now write ao =o, and substitute a 8 _ 1 for a,, when s>o, we obtain d d aO da l +al da 2 +a2 da ï¿½....+an_2dan_1 which is the form of SZ for a binary (n- Henceby merely diminishing each suffix in a seminvariant by unity, we obtain another seminvariant of the same degree, and of weight w-8, appertaining to the (n-I) ic. Also, if we increase each suffix in a seminvariant, we obtain terms, free from a 0, of some seminvariant of degree 8 and weight w+8.

• Hence, excluding ao, we may, in partition notation, write down the fundamental solutions of the equation, viz.

• (2), (3), (4),...(n), and say that with ao, we have an algebraically complete system.

• Observe that, if we subject any symmetric function the diminishing process, it becomes ao 1 - P2 (p2p3...)ï¿½ Next consider the solutions of 0=o o which are of degree 0 and weight w.

• When the proper degree 0 is < w a factor ao -e must be of course understood.

• 1 And The Actual Forms For The First Three Weights Are 1 Aobzo, (Ao B 1 A 1 B O) Bo, (A O B 2 A 1 2 0 Bo, Ao(B2, 3 A1B2 A2B1 A O (B L B 2 3B O B 3) A I (B 2 1 2B 0 B 2); Amongst These Forms Are Included All The Asyzygetic Forms Of Degrees 1, 1, Multiplied By Bo, And Also All The Perpetuants Of The Second Binary Form Multiplied By Ao; Hence We Have To Subtract From The 2 Generating Function 1Z And 1 Z Z2, And Obtain The Generating Function Of Perpetuants Of Degrees I, 2.

• 1 Z2' The First Perpetuant Is The Last Seminvariant Written, Viz.: A O (B O B 2 3B O B 3) A L (Bi 2B0B2), Or, In Partition Notation, Ao(21) B (1)A(2)B; And, In This Form, It Is At Once Seen To Satisfy The Partial Differential Equation.

• (0')B Denotes A Seminvariant, If 0, 0', Be Neither Of Them Unity, For, After Operation, The Terms Destroy One Another In Pairs: When 0, Must Be Taken To Denote Ao And So For 0'.

• For The Case In Hand, From The Simplest Perpetuant Of Degrees I, 2, We Derive The Perpetuants Of Weight W, Ao(21W 2)B A1(21"R 3) I A2(21" 4)B ..ï¿½ Man 2(2)B, Ao(221W 4(B Al(221W 5)B A2 (221 " S)B ...

• Maw 4(22)B, Ao(231W 9B A, (221" 7)B A2(231W 8)B ...

• For w = i the form is A i ai+Bib i, which we may write aob l -albo = ao(I) b -(I)abo; the remaining perpetuants, enumerated by z I - 2' have been set forth above.

• (14) a (4322) b - (13) a (432 2 1) b + (12)a (432212) b - (1) a (432213)b +ao(432214)b, and thence the general form (1 A i + 4) a (4ï¿½4 + 1.3 ï¿½ 3 + 121 1, 2 + 2) b - ... ?ao(4ï¿½4+13ï¿½a+l2ï¿½2+21 Al+4)b, due to the generating function 2 15 1 -z.

• This is the fundamental system; we may, if we choose, replace (ab) 2 by ab =a, 2, +2a1+a2 since the identity a a b b - a, = (ab) 2 shows the syzygetic relation (74+a 2) 2 - (ao +2a +ï¿½2) = 2(aoa2 - ai).

• Another example of a sequence is afforded by the successive convergents to a continued fraction of the form ao+ I I, al+ a2+ï¿½ where ao,a 1, a 2, ...

• Accordingly, the optical distance from AoBo to A is represented by f (A +S/c)ds, the integration being along the original path Ao.

• A 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)').

• The grating at A and the eye-piece at 0 are rigidly attached to a bar AO, whose ends rest on carriages, moving on rails OQ, AQ at right angles to each other.

• = constant, _ ff 00 NdA N BA-AA X - JA (a' +X) (b 2 +A)P - abc' a2 -b2 ' and at the surface A = o, I I N Bo-A 0 N I R - (a2+b2) abc a 2 -b 2 abc a2b2 I /b 2 N = R I /b2 - I /a2 abc I 1 I Bo - AO' a 2 b 2 - a2 b2 a 2 b2 = R (a 2 - b 2) /(a 22 + /b2) 2 - r (B o - Ao) U Bo+Co - B I - CI' Since - Ux is the velocity function for the liquid W' filling the ellipsoid A = o, and moving bodily with it, the effective inertia of the liquid in the interspace is Ao+B1+C1 Bo+Co - B1 - C, If the ellipsoid is of revolution, with b=c, - 2 XBo - - C BI' and the Stokes' current function 4, can be written down (I) is (5) (7) (6) The velocity function of the liquid inside the ellipsoid A=o due to the same angular velocity will be = Rxy (a2 - b2)/(a2 + b2), (7) and on the surface outside _ N Bo -Ao c1)0xy abc 2 62' so that the ratio of the exterior and interior value of at the surface is ?o= Bo-Ao (9) 4)1 (a 2 -6 2)/(a2 + b) - (Bo - Ao)' and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid X = o.

• This was the tribe of Kinda under the sway of the family of AO ul Murar, who came from the south.

• This can be done by placing at B an equal negative point-charge -q in the place which would be occupied by the optical image of A if PO were a mirror, that is, let -q be placed at B, so that the distance BO is equal to the distance AO, whilst AOB is at right angles to PO.

• The areas of the sides for which 0 and x=xo+2h, and of the section by the plane x=xo+h, may be found by Simpson's second formula; call these Ao and A2, and Al.

• x 2,2 - k 2 a2,o - h 2 a6,2 +ia4h2k2ao,0 5 2x - 1100-1,1 5 2x h2x k2x3,o+41-gh2k°0-1,o where ao,o is the total volume of the briquette.

• Also since dx has been stretched to +dy p&,(dx +dy) =po&odx or p&'(I +dy/dx) = (29) Substituting from (28) in (27) Y&a + P(2)U 2 (I + dy (3) 2 = p oc?oU 2, 0) and substituting from (29) in (30) Y&ao dx + pocZoU 2 + dx) = p owoU 2, (31) whence Yc = powoU2, or U2 = Y/ p, (32) where now p is the normal density of the rod.

• M?[aughfin .. ?Mor ° D Ao,;?:,K Frederick.

• Some of the political leaders escaped over the frontier - among them Prof. Thomas Garrigue Masaryk and Dr. Eduard Benes, who were subsequently to lead a success :3,500,000 English Miles 1, 0 ao Kilometres Czechoslovakia Frontier 1921 poilow stria-Hungary 191C411.

• Thus, if a ray of white light AO (fig.

• The last verse, with its two-fold greeting (6 14:nos, uera Tou 7rveuµar6s co y, 7) x6.pcs AO' upL ' v), shows unconsciously but plainly that, while the epistle professes to be a private letter to Timothy, it is in reality addressed to a wider circle, like 1 Tim.

• ?P ?Op aMo ?a, n ao ?

• b, buccal mass; m, retractor muscles of the buccal mass; ov, ovary; od, oviduct; i, coils of intestines; ao, aorta; c', left auricle; c, ventricle.

• The English Miles ro ao 30 Albae -tbae Capitals of Counties County Boundaries Railways --i--- Canals Marshes  ?_ - ' 1s3nge ?

• or negative according as they lie to the right or left of AO.

• of the vectors AB and AC (or BD), and of their sum AD, on a line perpendicular to AO, this is obvious.

• Thus a, negative rotation about OA may be regarded as a positive rotation about OA, the prolongation of AO.

• If)~, u, 1 be the components of momentum, we have X = AO, aT

• 131 AO represents an upright axis or spindle; B a weightcalled a bob, suspended by rod OB from a horizontal axis at 0, carried by the vertical axis.

• Then, the crank standing at any angle with the line of stroke, draw LP at right angles to the connecting rod, PN at right angles to the line of stroke OB and NA at right angles to the connecting rod; then AO is the acceleration of the point B to the scale on which KO represents the acceleration of the point K.

• ao `v„ irehar s.

• Gomez Herrero, Diccionario-gwta legislativo espanol (5 vols., Madrid, 1901-1903); Estadistica de la administracidn de justicia en to criminal durante; Boleiin mensual de estadistica demo grdfica-sanitaria de la peninsula y islas adjacenies (Madrid, monthly); Estado general de la armada para el ao; C. Fernandez Duro, Armada espanola desde la union de los reinos de Castilla y de Leon (9 vols., Madrid, 1895I 903); Boletin oficial del minisierio de marina.

• nn, mn, nj, and sometimes to initial n: ao (a nn u m), dano (d a m n u m), nudo (n 0 d u m).

• dontinae and Desmognathinae), on the vomers, Ao, Left aortic Pterygoids and parasphenoid (some Pelobates), arch.

• aviation brigade elements deploy to the AO.

• The ROE will be prepared and issued by a higher headquarters before any of the aviation brigade elements deploy to the AO.

• Tu Mu's somewhat spiteful charge against Ts`ao Kung has already been considered elsewhere.

• Rev. (1889); Meliarakes, IvTopta Tou f ao X Iov NLeaLas Kal SeaIIor6Tov T1] S 'Hiretpov, pp. 539-627 (Athens, 1898).

• 000 English Miles ao 20 30 an 5 c+ ?

• The wire being paid out without slack measures the actual distance and speed over the ground, and the engineer in charge is relieved of all anxiety in estimating the depth from the scattered soundings of the preliminary survey, or in calculating the retarding strain required to produce the specified slack, since the brakesman merely has to follow the indications of the instrument and regulate the strain so as to keep the pointer at the figure required - an easy task, seeing that the ratio of speed of wire and cable is not affected by the motion of the ship, whatever be the state of the sea, whereas the will I',/ OW= o a ' 30 30 ao.

• Every plant is constrained to carry Out its functions of germination, growth, nutrition, reproduction, &c., between certain limits of temperature, and somewhere between the extremes of these limits each function finds ao optimum temperature at which the working of the living machinery is at its best, and, other things being equal, any great departure from this may induce pathological conditions; and many disasters are due to the failure to provide such suitable temperaturese.g.

• ao; Dan.

• p. 256 (19 Ao).

• We have to multiply a01; -alas+a2 by ao, -aif32+a2 and we obtain ao (3 - aoal(f31N2 +01133) +aoa2(SI+13) -i-a?31a2 - aIa2(31 + 02) + al, 131+02 = b, 131132 = b t'i +s2 = 2bob2, and clearing of fractions R 1,5 = (a o b 2 - a2 b o) 2 + (a i b o - aobi) (aib2 - a2b1).

• Thus to obtain the resultant of aox 3 +a i x 2 +a 2 x+a 3, 4, =box2+bix+b2 we assume the identity (Box+Bi)(aox 3 +aix 2 +a2x+a3) = (Aox 2 +Aix+ A 2) (box2+bix+b2), and derive the linear equations Boa Ã‚° - Ac b o = 0, Boa t +B i ao - A 0 b 1 - A 1 bo =0, Boa t +B 1 a 1 - A0b2 - A1b1-A2bÃ‚° = 0, Boa3+Bla2 - A l b 2 -A 2 b 1 =0, B 1 a 3 - A 2 b 2 =0, = = (y l, y2,...ynl `x1, x2,...xnl for brevity.

• and by elimination we obtain the resultant ao 0 bo 0 0 al ao b1 bo 0 a 2 a i b 2 b 1 bo a numerical factor being disregarded.

• Put (aox 3 -}-a l x 2 +a 2 x +a 3) (box' +b1x'+b2) - (aox'3+aix'2+a2x'+a3) (box' + bix + b2) = 0; after division by x-x the three equations are formed aobcx 2 = aobix+aob2 =0, aobix 2 + (aob2+a1b1-a2bo) x +alb2 -a3bo = 0, aob2x 2 +(a02-a3bo)x+a2b2-a3b1 =0 and thence the resultant aobo ao aob2 aob 1 aob2+a1b1-a2bo alb2-a3b0 aob 2 a1b2 - a 3 bo a2b2 - a3b1 which is a symmetrical determinant.

• 1 X discriminant of f = ao X disct.

• This arises from the circumstance that the general operator Ao,a0aa1 + ialaa2 + 2a2 a 3 +...

• is transformed into the operator d 1 by the substitution (ac, al, a2, Ã¯¿½Ã¯¿½Ã¯¿½as, Ã¯¿½Ã¯¿½Ã¯¿½) _ (ao, Xoai, X 6 X i a 2, Ã¯¿½Ã¯¿½Ã¯¿½, XcX1..%s_las,Ã¯¿½Ã¯¿½Ã¯¿½), so that the theory of the general operator is coincident with that of the particular operator d1.

• For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x-h for x causes ao, a1, a 2 a3,...

• to become respectively ao, ai+hao, a2+2ha1, a 3 +3ha 2, ...

• ao -ialaan+l m !

• ao (m -2) m !

• being formed, the expansion being carried out, an operand ao or bo or co ...

• F(a ' a ' a, ...a) =r A F(ao, a1, a2,Ã¯¿½Ã¯¿½Ã¯¿½an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation.

• ;51, 2) = r F(ao, al, a2,...; xi, x2), the function F(a 01 a 1, a 2, ...

• Instead of a single quantic we may have several f(ao, a1, a2...; x1, x2), 4 (b o, b1, b2,...; x1, x2), ...

• For the substitution rr xl =A 11 +1 2 12, 52=A21+Ã¯¿½2E2, of modulus A1 Ã¯¿½i = (AlÃ¯¿½.2-A2Ã¯¿½1) = (AM), A 2 Ã¯¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lÃ¯¿½l +ai(A1/.22+A2Ã¯¿½1) +7,2X2/22, A2 = aoÃ¯¿½l +2a1Ã¯¿½1/Ã¯¿½2 +a 2Ã¯¿½2 Ã¯¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?

• A 1, A2 Ã¯¿½ Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (alÃ¯¿½l +a2Ã¯¿½2) = aAaÃ¯¿½, A 2 = (alÃ¯¿½l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a Ã¯¿½ = a1/Ã¯¿½1 + a21=2 respectively.

• By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1Ã¯¿½1 +a2m2) = aa a Ã¯¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alÃ¯¿½1+a2Ã¯¿½2) = a A Ã¯¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a Ã¯¿½ respectively.

• For, if c(ao i ...x l, x 2) be a covariant of order e appertaining to a quantic of order n, t (T.

• 1 2) = (A /-?) ' (ao,...

• The Binary Quartic.-The fundamental system consists of five forms ax=f; (f,f')2=(ab) 2axbx=Ax; (f,f')4=(ab) 4= 2; (f, 0)1= (ao) azsi = (ab) 2 (cb) a:b x c5 =1; (f 4) 4 = (as) 4 = (6) 2 00 2 (ca) 2 = j, viz.

• l aa k -x 2 d d- = 0; Z(nk)ak+l adk - x ldd2=0; or in the form d d 52-x 2(7 =0, O - x1ax2 = 0; where 0 = ao d a l + 2a 1 -?...+na,,_id an, 0 = nal dao -?

• One advantage we have obtained is that, if we now write ao =o, and substitute a 8 _ 1 for a,, when s>o, we obtain d d aO da l +al da 2 +a2 da Ã¯¿½....+an_2dan_1 which is the form of SZ for a binary (n- Henceby merely diminishing each suffix in a seminvariant by unity, we obtain another seminvariant of the same degree, and of weight w-8, appertaining to the (n-I) ic. Also, if we increase each suffix in a seminvariant, we obtain terms, free from a 0, of some seminvariant of degree 8 and weight w+8.

• Hence, excluding ao, we may, in partition notation, write down the fundamental solutions of the equation, viz.

• (2), (3), (4),...(n), and say that with ao, we have an algebraically complete system.

• Observe that, if we subject any symmetric function the diminishing process, it becomes ao 1 - P2 (p2p3...)Ã¯¿½ Next consider the solutions of 0=o o which are of degree 0 and weight w.

• When the proper degree 0 is < w a factor ao -e must be of course understood.

• For Two Forms The Seminvariants Of Degrees I, I Are Enumerated By 1 Z, And The Only One Which Is Reducible Is Ao 0 Of Weight Zero; 1 Hence The Perpetuants Of Degrees I, I Are Enumerated By 11 1 Ã¯¿½ Z 1Zz' And The Series Is Evidently A O B 1 Aibo, A 0 B 2 A B A2Bo, A O B 3 A L B 2 A 2 B 1 A3Bo, One For Each Of The Weights I, 2, 3,..Ad Infin.

• 1 And The Actual Forms For The First Three Weights Are 1 Aobzo, (Ao B 1 A 1 B O) Bo, (A O B 2 A 1 2 0 Bo, Ao(B2, 3 A1B2 A2B1 A O (B L B 2 3B O B 3) A I (B 2 1 2B 0 B 2); Amongst These Forms Are Included All The Asyzygetic Forms Of Degrees 1, 1, Multiplied By Bo, And Also All The Perpetuants Of The Second Binary Form Multiplied By Ao; Hence We Have To Subtract From The 2 Generating Function 1Z And 1 Z Z2, And Obtain The Generating Function Of Perpetuants Of Degrees I, 2.

• 1 Z2' The First Perpetuant Is The Last Seminvariant Written, Viz.: A O (B O B 2 3B O B 3) A L (Bi 2B0B2), Or, In Partition Notation, Ao(21) B (1)A(2)B; And, In This Form, It Is At Once Seen To Satisfy The Partial Differential Equation.

• (0')B Denotes A Seminvariant, If 0, 0', Be Neither Of Them Unity, For, After Operation, The Terms Destroy One Another In Pairs: When 0, Must Be Taken To Denote Ao And So For 0'.

• For The Case In Hand, From The Simplest Perpetuant Of Degrees I, 2, We Derive The Perpetuants Of Weight W, Ao(21W 2)B A1(21"R 3) I A2(21" 4)B ..Ã¯¿½ Man 2(2)B, Ao(221W 4(B Al(221W 5)B A2 (221 " S)B ...

• Maw 4(22)B, Ao(231W 9B A, (221" 7)B A2(231W 8)B ...

• For w = i the form is A i ai+Bib i, which we may write aob l -albo = ao(I) b -(I)abo; the remaining perpetuants, enumerated by z I - 2' have been set forth above.

• (14) a (4322) b - (13) a (432 2 1) b + (12)a (432212) b - (1) a (432213)b +ao(432214)b, and thence the general form (1 A i + 4) a (4Ã¯¿½4 + 1.3 Ã¯¿½ 3 + 121 1, 2 + 2) b - ... ?ao(4Ã¯¿½4+13Ã¯¿½a+l2Ã¯¿½2+21 Al+4)b, due to the generating function 2 15 1 -z.

• For the quadratic aoxi +2a i x i x 2 +a 2 x, we have (i.) ax = 7/1x1+2aixix2-I-7/24, (ii.) xx=xi+xzi (ab) 2 =2(aoa2 - ai), a a = a o+712, _ (v.) (xa)ax= i'?- (a2 - ao)xix2 - aix2.

• This is the fundamental system; we may, if we choose, replace (ab) 2 by ab =a, 2, +2a1+a2 since the identity a a b b - a, = (ab) 2 shows the syzygetic relation (74+a 2) 2 - (ao +2a +Ã¯¿½2) = 2(aoa2 - ai).

• oe 000000 0 o co e io??,a _e ` ao O o' 3 4;Ã‚° :o o moo-:„..„:0-„,-----0 ?O 1/ // /O i Q n.

• Another example of a sequence is afforded by the successive convergents to a continued fraction of the form ao+ I I, al+ a2+Ã¯¿½ where ao,a 1, a 2, ...

• Accordingly, the optical distance from AoBo to A is represented by f (A +S/c)ds, the integration being along the original path Ao.

• A 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)').

• The grating at A and the eye-piece at 0 are rigidly attached to a bar AO, whose ends rest on carriages, moving on rails OQ, AQ at right angles to each other.

• Similarly, the inertia parallel to Oy and Oz is NW' - 1 B W', B C (b2 +-X, c 2 ab and A +C abc/ZP, Ao For a sphere a=b=c, Ao= Bo=Co =, 'a' = Q = = z, (9) U from (II), (16) so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration W - W', g (30) W+ aW Thus a spherical air bubble, in which W/W' is insensible, will begin to rise in water with acceleration 2g.

• = constant, _ ff 00 NdA N BA-AA X - JA (a' +X) (b 2 +A)P - abc' a2 -b2 ' and at the surface A = o, I I N Bo-A 0 N I R - (a2+b2) abc a 2 -b 2 abc a2b2 I /b 2 N = R I /b2 - I /a2 abc I 1 I Bo - AO' a 2 b 2 - a2 b2 a 2 b2 = R (a 2 - b 2) /(a 22 + /b2) 2 - r (B o - Ao) U Bo+Co - B I - CI' Since - Ux is the velocity function for the liquid W' filling the ellipsoid A = o, and moving bodily with it, the effective inertia of the liquid in the interspace is Ao+B1+C1 Bo+Co - B1 - C, If the ellipsoid is of revolution, with b=c, - 2 XBo - - C BI' and the Stokes' current function 4, can be written down (I) is (5) (7) (6) The velocity function of the liquid inside the ellipsoid A=o due to the same angular velocity will be = Rxy (a2 - b2)/(a2 + b2), (7) and on the surface outside _ N Bo -Ao c1)0xy abc 2 62' so that the ratio of the exterior and interior value of at the surface is ?o= Bo-Ao (9) 4)1 (a 2 -6 2)/(a2 + b) - (Bo - Ao)' and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid X = o.

• This was the tribe of Kinda under the sway of the family of AO ul Murar, who came from the south.

• This can be done by placing at B an equal negative point-charge -q in the place which would be occupied by the optical image of A if PO were a mirror, that is, let -q be placed at B, so that the distance BO is equal to the distance AO, whilst AOB is at right angles to PO.

• The areas of the sides for which 0 and x=xo+2h, and of the section by the plane x=xo+h, may be found by Simpson's second formula; call these Ao and A2, and Al.

• x 2,2 - k 2 a2,o - h 2 a6,2 +ia4h2k2ao,0 5 2x - 1100-1,1 5 2x h2x k2x3,o+41-gh2kÃ‚°0-1,o where ao,o is the total volume of the briquette.

• Also since dx has been stretched to +dy p&,(dx +dy) =po&odx or p&'(I +dy/dx) = (29) Substituting from (28) in (27) Y&a + P(2)U 2 (I + dy (3) 2 = p oc?oU 2, 0) and substituting from (29) in (30) Y&ao dx + pocZoU 2 + dx) = p owoU 2, (31) whence Yc = powoU2, or U2 = Y/ p, (32) where now p is the normal density of the rod.

• M?[aughfin .. ?Mor Ã‚° D Ao,;?:,K Frederick.

• Some of the political leaders escaped over the frontier - among them Prof. Thomas Garrigue Masaryk and Dr. Eduard Benes, who were subsequently to lead a success :3,500,000 English Miles 1, 0 ao Kilometres Czechoslovakia Frontier 1921 poilow stria-Hungary 191C411.

• Thus, if a ray of white light AO (fig.

• The last verse, with its two-fold greeting (6 14:nos, uera Tou 7rveuµar6s co y, 7) x6.pcs AO' upL ' v), shows unconsciously but plainly that, while the epistle professes to be a private letter to Timothy, it is in reality addressed to a wider circle, like 1 Tim.

• It distinguishes prudence (4poinacs) and wisdom (ao,ia) as the respective virtues of deliberative and scientific reason; and on the whole its account of prudence (cf.

• ?P ?Op aMo ?a, n ao ?

• b, buccal mass; m, retractor muscles of the buccal mass; ov, ovary; od, oviduct; i, coils of intestines; ao, aorta; c', left auricle; c, ventricle.

• The English Miles ro ao 30 Albae -tbae Capitals of Counties County Boundaries Railways --i--- Canals Marshes  ?_ - ' 1s3nge ?

• or negative according as they lie to the right or left of AO.

• of the vectors AB and AC (or BD), and of their sum AD, on a line perpendicular to AO, this is obvious.

• Thus a, negative rotation about OA may be regarded as a positive rotation about OA, the prolongation of AO.

• If)~, u, 1 be the components of momentum, we have X = AO, aT

• 131 AO represents an upright axis or spindle; B a weightcalled a bob, suspended by rod OB from a horizontal axis at 0, carried by the vertical axis.

• Then, the crank standing at any angle with the line of stroke, draw LP at right angles to the connecting rod, PN at right angles to the line of stroke OB and NA at right angles to the connecting rod; then AO is the acceleration of the point B to the scale on which KO represents the acceleration of the point K.

• ao `v„ irehar s.

• Gomez Herrero, Diccionario-gwta legislativo espanol (5 vols., Madrid, 1901-1903); Estadistica de la administracidn de justicia en to criminal durante; Boleiin mensual de estadistica demo grdfica-sanitaria de la peninsula y islas adjacenies (Madrid, monthly); Estado general de la armada para el ao; C. Fernandez Duro, Armada espanola desde la union de los reinos de Castilla y de Leon (9 vols., Madrid, 1895I 903); Boletin oficial del minisierio de marina.

• nn, mn, nj, and sometimes to initial n: ao (a nn u m), dano (d a m n u m), nudo (n 0 d u m).

• dontinae and Desmognathinae), on the vomers, Ao, Left aortic Pterygoids and parasphenoid (some Pelobates), arch.

• Tu Mu 's somewhat spiteful charge against Ts`ao Kung has already been considered elsewhere.

• Science Drivers for ASMs Currently implemented astronomical AO systems operate like auxiliary instruments separate from the main telescope optics.

• The AO acts as a fair and unbiased referee when people feel they have been badly treated.

• While the publisher was instructed by ESRB to request that retailers stop selling the Mature-labeled game until an AO sticker was available, the game has been spotted on retail shelves at stores including Fry's Electronics.

• Since many major retailers will not carry games with an AO rating, this can adversely affect a game's sales.

• After the disclosure that a hack could expose an explicit consensual sex scene in Grand Theft Auto: San Andreas, the game's rating was changed to AO for Adults Only from its previous Mature rating status.

• Instead, the ESRB gave the game an AO (Adults Only) rating because after a lengthy investigation, the game was found to have some hidden content that involved nudity and prostitution.

• Retailers immediately pulled the AO version from the shelves and replaced it with the toned down Mature version.

• Developers are apprehensive about putting video game nudity into their creations because it is proven that games with the Adults Only (AO) rating are poor performers.

• The current list of AO games is at 30 and you can view this list at Moby Games.

• A quick perusal of the titles and descriptions might make you realize that they probably didn't sell very well not because of the AO rating, but because they simply stunk.

• If you notice, the last AO game that was released was in 2005.

• There is no shortage of nudity and because the character models are so realistic, it's no surprise this game received its AO rating.

• Sure there were polygon babes riding BMX bikes, but the clips are what caused the controversy and the eventual AO rating.

• AO - Adults Only - Aimed at adults over the age of 17 only, these games may include prolonged scenes of violence, graphic sexual content and nudity.

• AO for Adults Only: No explanation needed.

• AO - Adults Only: These adult video games may be pornographic in nature and may contain excessive blood and gore.

• Very few games receive this rating, because Sony, Nintendo, and Microsoft all prohibit the sale of AO games on their respective consoles.

• If you loved Switched-on-Schoolhouse, (AO's computer-based learning curriculum) you may well love Monarch even more.

• Similar to motion picture ratings, the ESRB ratings range from EC (Early Childhood - appropriate for ages three and up) to AO (Adults Only - a rating that is rarely seen and not carried by most retailers).