# A-2 sentence example

a-2
• The values of a 2 at the various stations differ comparatively little, and show but little seasonal change.
• The 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).
• The equation then becomes a 2 /V = k, or a = A / Vk, so that the molecular conductivity is proportional to the square root of the dilution.
• Let there be given n 2 quantities all a,2 a13 ï¿½ï¿½ï¿½ aln a21 a22 a23 ï¿½ï¿½ï¿½ a2n a3, a32 a33 ï¿½ï¿½ï¿½ a3n and an, an 3 ï¿½ï¿½ï¿½ ann and form from them a product of n quantities ala a2 0 a37 ...
• ï¿½ï¿½ a n1 - 1 a12 a 22 a32 ï¿½ï¿½ï¿½ an2 0 a13 a 23 a33 ï¿½ï¿½ï¿½ an3 0 a3n ï¿½ï¿½ ï¿½ a nn 0 0 0 ...
• If we write (I +a i x) (I a 2 x) ...
• The function Zap 1 a 2 P2 ...an n being as above denoted by a partition of the weight, viz.
• 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,...
• It is thus possible to study simultaneously all the theories which depend upon operations of the group. Symbolic Representation of Symmetric Functions.-Denote the s 8 s elementar symmetric function a s by al a 2 a3 ...at pleasure; then, Y y si,, si,...
• If we restrict ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the symbolic representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write n-1 n-2 2 2n-3 3 a1a2 =a l a 2, a 1 a 2 = a 1 a2 while the same product of umbrae arises from n n-3 3 2n-3 3 aoa 3 = a l .a a 2 = a a 2 .
• We consider the quantic to have any n number of equivalent representations a- b n -c n So that a 1 -k a 2 = b l -k b 2 - c 1 -k c 2 = ...
• We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism.
• 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.
• = (A11+A22)n by the substitutions 51 = A l, E1+ï¿½1 2, 52 = A2E1+ï¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = ï¿½la1 +ï¿½2a2ï¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).
• For this reason the umbrae A1, A 2 are said to be contragredient to xi, x 2.
• As between the original and transformed quantic we have the umbral relations A1 = A1a1 d-A2a2, A2 = /21a1+/22a2, and for a second form B1 =A 1 b 1+ A 2 b 2, B 2 =/21bl +ï¿½2b2ï¿½ The original forms are ax, bi, and we may regard them either as different forms or as equivalent representations of the same form.
• For the quartic (ab) 4 = (aib2-a2b,) alb2 -4a7a2blb2+64a2 bib2 - 4a 1 a 2 b 7 b 2 + a a b i = a,a 4 - 4ca,a 3 +6a2 - 4a3a3+ aoa4 = 2(a 0 a 4 - 4a1a3 +e3a2), one of the well-known invariants of the quartic.
• First observe that with f x =a: = b z = ï¿½ï¿½ï¿½,f1 = a l a z ', f 2 = a 2 az-', f x =f,x i +f 2 x i, we find (ab) - (a f) bx - (b f) ax.
• The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.
• Again, if 0 is uneven =20+I, the condition is a 1 a 2 ...cr 241 II(a 1 +a 2)II(cr 1 +a 2 +(73)...II(a 1 +a 2 +...+ac) =0; and the degree, in the quantities a, is 20+1 + (42+1) +(21) ï¿½...-F(254)ï¿½1) =22°-1= 2e-1-1 Hence the lowest weight of a perpetuant is 2 0 - 1 -1, when 0 is >2.
• To represent the simplest perpetuant, of weight 7, we may take as base either A2B 1 B 2 or A l A 2 B2, and since Ai+Bi =o the former is equivalent to A 2 ArB 2 and the latter to A 2 B i B2; so that we have, (1 -f-aix) (1 + a2x).
• We will choose from the forms in such manner that the product of letters A is either a power of A i, or does not contain A i; this rule leaves us with A2B 1 B 2 and A 2 B,Bs; of these forms we will choose that one which in letters B is earliest in ascending dictionary order; this is A2B 1 B 21 and our earliest perpetuant is (22)a(21)b - (221)a(2)b, and thence the general form enumerated by the generating function Z7 is (1-z)(1 - z2)2 (2 A2+2) a (2ï¿½2 +1 1ï¿½1 +1) b - (2 A2+2 1)a (2 M2+1 1, ai)b ...
• The calculation results in -A,113 B2Bi+2A2B3B2Bi-AzB3BaBi+A4B3B1-2AlB3B2B1 -MB2B2131+MB33231+A213MB1 + A 2 B a Bi -2A2B3B2B +A2B1B1=0.
• The session was interrupted by the outbreak of the Austro-Prussian War, but not before a 2 Transylvania, Croatio-Slavonia with Fiume and the Temes Banat were separated from the kingdom and provided with local governments.
• These formulae are of two kinds: - (a) the general properties, such as m(a+b) = ma+mb, on which algebra is based, and (b) particular formulae such as (x - a) (x+a) = x 2 - a 2 .
• The theory may be extended to the cases of p= i and p = o; so that a 3 means a.a.a.1, a 2 means a.a.i, a 1 means a.i, and a° means I (there being then none of the multipliers a).
• Similarly the equalities 99 X I o I = 9999 = wow - I 98 X 102 = 999 6 = moo() - 4 97 X 10 3 =9991 =1 0000 - 9 lead up to (A - a) (A+a) = A 2 - a 2.
• These, with (A - a)2= A 2 -2Aa+a 2, are the most important in elementary work.
• (iii.) By writing (A+a) 2 = A 2 + 2Aa+a 2 in the form (A+a)2= A 2 +(2A+a)a, we obtain the rule for extracting the square root in arithmetic.
• The coefficient of A 2 a 3 in the expansion of (A+a) 5 is then the number of terms such as ABcde, AbcDe, AbCde, ...
• The first term is Abcde, in which all the letters are large; and the coefficient of A 2 a 3 is therefore the number of terms which can be obtained from Abcde by changing three, and three only, of the large letters into small ones.
• If we divide the sum of x 2 and a 2 by the sum of x and a, we get a quotient x - a and remainder 2a 2, or a quotient a - x and remainder 2x 2, according to the order in which we work.
• A diminution of X thus leads to a simple proportional shrinkage of the diffraction pattern, attended by an augmentation of brilliancy in proportion to A-2.
• Analytically expressed ff+ co x I 2 d dn=ff dxdy= A (9) We have seen that Io (the intensity at the focal point) was equal to A 2 /X 2 f2.
• In the case of a single lens of glass with the most favourable curvatures, Sf is about equal to a 2 f, so that a 4 must not exceed off.
• In the limiting case in which the medium is regarded as absolutely incompressible S vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a 2 3 by a new function of the co-ordinates.
• (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.
• Then 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).
• When 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?
• Then, if the outside cylinder is free to move - 2a2 2 X 1 = 0, T T = b2 a2, 7rpa 2 Ub 2+ a 2.
• Thus, for example, with = 4Uy 2 (r 2 a 2 -I), r2 = x2 +y 2, (13) for the space inside the sphere r=a, compared with the value of, i' in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.
• From 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.
• Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, px _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p d = ds; (8) Thence d?
• Between two concentric spheres, with a 2 +A = r 2, a2+A1=a12, A=B =C =a3/3r3, a 3 a 3 a3 _ a3 Cb _1U x r 3 a13 Y'=2 Uy2 r3 3 a13 .
• Thus for m =2, the spheres are orthogonal, and it can be verified that a13 a2 3 aY3 i f /' = ZU (I - 13 - 7.2 3 + 3) ' (8) where a l, a2, a =a l a 2 /J (a 1 2 +a 2 2) is the radius of the spheres and their circle of intersection, and r 1, r 2, r the distances of a point from their centres.
• The corresponding expression for two orthogonal cylinders will be With a 2 = co, these reduce to / y /, = Uy (I ra 2 p22 +-C24)..
• The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= X is made in a similar manner, by putting 4 = x y (x+ 11), (io) and the ratio of the effective angular inertia in (9) is changed to 2 (B0-A0) (B 1A1) +.a12 - a 2 +b 2 a b1c1 a -b -b12 abc (Bo-Ao)+(B1-A1) a 2 + b 2 a1 2 + b1 2 alblcl Make c= CO for confocal elliptic cylinders; and then _, 2 A? ?
• C - a 2 - 2 b2 +A, = and then as above in § 31, with a= c ch a, b=c sh a, a =-1 (a 2 +X) =c ch al, b1= c sh a (13) the ratio in (II) agrees with § 31 (6).
• - 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.
• We 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.
• Try = o to be a parabola is lbc+mca+nab = o, and the conic for which the triangle of reference is self-conjugate la 2 +143 2 +n7 2 =o is a 2 inn--+b 2 nl+c 2 lm=o.
• In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.
• The notation employed by English writers for the general continued fraction is al b2 b3 b4 a 2 "' Continental writers frequently use the notation a 1 ?
• Let the fraction be a l a 2 al +...
• 'a ' Let un = nr+m; then u n = 1 a 2 a r leading to an - km equation of the form Au n u n _,+ Bu„ +Cu n _,+D =o, where A,B,C,D are independent of n, which is readily solved.
• If a i, a 2 represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area 21ra l a fZ '(z)dz, (30) or 27ra l 02 0(z), if we write f 4,(z)dz=0(z) (31) The work required to produce the separation in question is thus 2 7ru l a o 0 (z)dz; (32) and for the tension of a liquid of density a we have T = a f o 0 (z)dz.
• Resolving vertically we find that the weight of the liquid raised above the level must be equal to T(sin 0 2 - sin 01), and this is therefore equal to the area P 1 P 2 A 2 A 1 multiplied by gp. The form of the capillary surface is identical with that of the " elastic curve," or the curve formed by a uniform spring originally straight, when its ends are acted on by equal and T 2 opposite forces applied either to the ends themselves or to solid pieces attached to them.
• Hence in all cases except that in which the angles a l and a 2 are supplementary to each other, the force is attractive when a is small enough, but when cos a i and cos a 2 are of different signs, as when the liquid is raised by one plate, and depressed by the other, the first term may be so small that the repulsion indicated by the second term comes into play.
• F(ka), (2) where, as before, T is the superficial tension, p the density, and F is given by the following table: - The greatest value of F thus corresponds, not to a zero value of k 2 a 2, but approximately to k 2 a 2 = 4858, or to A = 4.508 X 2a.
• This curve is the envelope of a line of constant length, which moves so that its extremities are always on two fixed lines at right angles to each other, i.e.of the line xla+y//= I, with the condition a 2 + 1 3 2 = I/a, a constant.
• The Cartesian equation to the caustic produced by reflection at a circle of rays diverging from any point was obtained by Joseph Louis Lagrange; it may be expressed in theform 1(4,2_ a2) (x 2+ y2) - 2a 2 cx - a 2 c 2 1 3 = 2 7 a4c2y2 (x2 + y2 - c2)2, where a is the radius of the reflecting circle, and c the distance of the luminous point from the centre of the circle.
• Many bacteria are killed by a 2 solution of the alkaloid.
• Consider the traces these surfaces cut on any sphere r =a: we have de/de = 2e sin a cos e/{cos t - aR2 dR/de}, which is positive and has a maximum in the middle latitudes; so that, proceeding from the pole to the equator along any meridian, the angular velocity would continually in crease, at a rate which was greatest in the middle latitudes.
• Equation (I) becomes E/R= (I - b - c - M/R)/(i - a), and hence b+c +M /R is equal to or greater than unity when the load is self-sustained, and we thus obtain a relation between R and E in the form i - a/2 - c, which shows to a first approximation, that as c approaches unity a high efficiency is obtainable, while the self-sustaining power of the tackle is retained.
• The Hydrobromide A' acid results on boiling the A 2 acid on reduction with alkalis, or on eliminating hydroHEXAHYDRO bromic acid from i-brom-cyclo-hexane carboxylic acid-I.
• The A 2 acid is formed along with the A 4 acid by reducing phthalic acid with sodium amalgam in hot solutions.
• The A 2 acid is formed along with the A 4 (cis) acid by reducing isophthalic acid.
• When boiled with caustic soda it isomerizes to a mixture of the A 2.4 and A 2 '° dihydrophthalic acids.
• The acid is obtained by boiling the dihydrobromide of the A 2 '° acid with alcoholic potash or by continued boiling of the 2.6 acid with caustic soda.
• The 2' acid is formed when phthalic acid is reduced in the cold by sodium amalgam or by heating the A 2 ' 4 and A 3 " acids with caustic soda.
• (iii) (a+b) 2 = a 2 + 2ab+b 2 = a 2 + (2a+b) b.
• If the complete square root is a+b, the remainder after subtracting a 2 is (2a+b)b.
• -Fan-i common intersections, where the a, points are ordinary points, the a 2 points are double points, the a 3 points are triple points, &c., on each curve, we have the condition a i +4a 2 +9a 3 +...
• It may be added that the two equations together give a 2 +3a 3 ...
• =o, of the order m, = a l m l +a 2 m 2 + ..., composed of the curve P i = o taken a l times, the curve P2 = o taken a 2 times, &c. Instead of the equation P i Ct1 P 2 a2 ...
• In strictness the angle is dependent upon the frequency, but if the dispersion be weak relatively to the double refraction, the product sin 24 - a)sin 2Ni - (3) has sensibly the same value for all terms of the summation, and we may write I=cos 2 (1 3 - a)/a 2 - sin 2 (1 ' - a) sin 2 (t ' - a 2 sin 2 2 This formula contains the whole theory of the colours of crystalline plates in polarized light.
• Does the plane I fly have a 2 nd altimeter?
• For a 2 layer anastomosis: There are 4 parts instead of the 2 for the 1 layer method.
• Satin - being fostered in Somerset ywatts 30/06/06 9:05 Satin is a 2 year old ex breeding bitch yellow lab.
• Included is a 2 day Hopper Pass to the Disney® Parks and the spectacular bonfire and Fireworks celebrations.
• It has a terry outer with a 2 layer terry booster sewn in.
• What we want to talk about is how they managed to get 345 brake horsepower out of a 2 liter engine.
• Above right a 2 light casement in small roof gable in 1939.
• On 2nd floor a 3 light center opening casement either side a 2 light casement.
• We will be berthing at the Flamenco Marina which is on an island connected by a 2 mile causeway to the mainland.
• Some boys brought in a mare, her nine week old foal and a 2 year old colt for treatment and assessment.
• The price is based on 6 sharing a 2 bedroom condo.
• This is a 2 year project with several expeditions to the frozen continent.
• The drive to diversify the economy by attracting the tourist dollar has brought in a 2 tier system, divisive and visibly unfair.
• We can recognize the R-C frequency-sensitive network as being a 2 nd order bandpass filter.
• Each pack comes with 2 liners + 2 wing liners, both made from a 2 layer 100% cotton fleece construction.
• Comes with a 4m suction hose, remote control and a 2 year guarantee.
• These features normally include: A 2 way speech intercom system linked to the HSO.
• If you are tired, then go for 10 minutes jog If you are not tired, then go for a 2 mile jog.
• In each gable a 2 light casement the curving lintel ornamented with a surround of projecting brick keystones.
• I'm a 2 yr old lady Collie who is a bit of a control freak.
• Harvey, gentle rough coated lurcher Barb 26/02/06 2:21 Harvey is a 2 year old rough coated neutured lurcher.
• Tackle requirements: A 2 lb-plus test curve specimen rod, 12 lb mainline and a long hooklink of strong wire.
• Mais la WBS est super internationale, il y a 2 anglais Dans mon master!
• Thus HMT might involve a maximum of 20 additional days off-site spread over a 2 - 4 year period.
• Note 2: The use of MSS over a 2 or 3NT opener always shows slam interest.
• I do not think this would be possible with a 2 pack enamel paint finish, which is much more expensive.
• Results from a 2 year double blind placebo controlled trial.
• The campaign involved placing provocative and eye-catching posters on the sides of 75 Bristol busses during a 2 month period in 2003.
• To him we owe the symbols for and, a 2 and a 3, and the cube root sign.
• I wonder if this continual self-improvement is why we have a 2 party system.
• Your guests will probably enjoy a 2 to 5 course sit-down meal or a well-stocked buffet.
• The i310 has all the latest features, including a 2 megapixel camera with flash, microSD slot, document viewer and TV output.
• It may need to have a proximal stoma more often than a 2 layer closure.
• This in conjunction with a 2 speed bottom bracket gear made the six speed sunbeams possible.
• Mais la WBS est super internationale, IL y a 2 anglais dans mon master!
• Let there be given n 2 quantities all a,2 a13 Ã¯¿½Ã¯¿½Ã¯¿½ aln a21 a22 a23 Ã¯¿½Ã¯¿½Ã¯¿½ a2n a3, a32 a33 Ã¯¿½Ã¯¿½Ã¯¿½ a3n and an, an 3 Ã¯¿½Ã¯¿½Ã¯¿½ ann and form from them a product of n quantities ala a2 0 a37 ...
• Ã¯¿½Ã¯¿½ a n1 - 1 a12 a 22 a32 Ã¯¿½Ã¯¿½Ã¯¿½ an2 0 a13 a 23 a33 Ã¯¿½Ã¯¿½Ã¯¿½ an3 0 a3n Ã¯¿½Ã¯¿½ Ã¯¿½ a nn 0 0 0 ...
• If any symmetrical determinant vanish and be bordered as shown below all a12 a13 Al a12 a22 a23 A2 a13 a23 a33 A3 Al A2 A3 Ã¯¿½ it is a perfect square when considered as a function of A 11 A2, A3.
• It can be shown that the number 0 enumerates distributions of a certain nature defined by the partitions (,Ã¯¿½i,Ã¯¿½2...), (sT1sÃ‚°2...), 1212 an = a 1 a 2 a 3 ...
• For example, the theory of invariants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the equation aox n - (i) a i x n - 1 + (z) a 2 x n 2 - ...
• 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.
• = (A11+A22)n by the substitutions 51 = A l, E1+Ã¯¿½1 2, 52 = A2E1+Ã¯¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = Ã¯¿½la1 +Ã¯¿½2a2Ã¯¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).
• As between the original and transformed quantic we have the umbral relations A1 = A1a1 d-A2a2, A2 = /21a1+/22a2, and for a second form B1 =A 1 b 1+ A 2 b 2, B 2 =/21bl +Ã¯¿½2b2Ã¯¿½ The original forms are ax, bi, and we may regard them either as different forms or as equivalent representations of the same form.
• First observe that with f x =a: = b z = Ã¯¿½Ã¯¿½Ã¯¿½,f1 = a l a z ', f 2 = a 2 az-', f x =f,x i +f 2 x i, we find (ab) - (a f) bx - (b f) ax.
• Again, if 0 is uneven =20+I, the condition is a 1 a 2 ...cr 241 II(a 1 +a 2)II(cr 1 +a 2 +(73)...II(a 1 +a 2 +...+ac) =0; and the degree, in the quantities a, is 20+1 + (42+1) +(21) Ã¯¿½...-F(254)Ã¯¿½1) =22Ã‚°-1= 2e-1-1 Hence the lowest weight of a perpetuant is 2 0 - 1 -1, when 0 is >2.
• 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.
• A2B 1 B 2 gives (22) a (21) b - (221) a (2) b, and A i AgB 21 (2 2 I) a (2) b -(22)a(21) b; these two merely differ in sign; and similarly A 2 B 1 B2 yields (2)a(2 2 I) b -(21)a(22)b, and that due to A 1 A 2 B2 merely differs from it in sign.
• We will choose from the forms in such manner that the product of letters A is either a power of A i, or does not contain A i; this rule leaves us with A2B 1 B 2 and A 2 B,Bs; of these forms we will choose that one which in letters B is earliest in ascending dictionary order; this is A2B 1 B 21 and our earliest perpetuant is (22)a(21)b - (221)a(2)b, and thence the general form enumerated by the generating function Z7 is (1-z)(1 - z2)2 (2 A2+2) a (2Ã¯¿½2 +1 1Ã¯¿½1 +1) b - (2 A2+2 1)a (2 M2+1 1, ai)b ...
• The theory may be extended to the cases of p= i and p = o; so that a 3 means a.a.a.1, a 2 means a.a.i, a 1 means a.i, and aÃ‚° means I (there being then none of the multipliers a).
• (II) A single vortex in a circular cylinder of radius a at a distance c from the centre will move with the velocity due to an equal opposite image at a distance a 2 /c, and so describe a circle with velocity mc/(a 2 -c 2)in the periodic time 21r(a 2 -c 2)/m.
• With liquid of density p, this gives rise to a kinetic reaction to acceleration dU/dt, given by 7rp b 2 a 2 b l b d J = a 2 +b2 M' dU, if M' denotes the mass of liquid displaced by unit length of the cylinder r =b.
• For a discussion of this type of wave, u = dt = - U¢ cos (x - Ut), and Ã‚° 4, x Za2 / cos t (x - Ut) dx pu2ax=p = 2p7r 2 U 2 a 2 /X (12) The energy per cubic centimetre on the average is 2 pif2 U2a2 / A2 (13) and the energy passing per second through I sq.
• The circumcircle is thus seen Areal to be a 2 yz+b 2 zx+c 2 xy=o, with centre sin 2A, sin 2B, co sin 2C; the inscribed circle is A t (x cot ZA)+ (y cot 2B) nates.
• The general theorems which enabled him to do this, after a start had been made, are A2n = 11A„A ' n (Snell's Cyclom.), P 2A„A' n - 2A' „AZ, Gre o A 2 ” - A n +A2n or A' n +A2„ (g r1') where A „, A'„ are the areas of the inscribed and the circumscribed regular n-gons respectively.
• The forces acting on the portion of liquid P 1 P 2 A 2 A 1 are - first, the horizontal pressures, - pgy i and z pgy 2; second, the surface-tension T acting at P i and P2 in directions inclined 01 and 0 2 to the horizon.
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• The acid is obtained by boiling the dihydrobromide of the A 2 'Ã‚° acid with alcoholic potash or by continued boiling of the 2.6 acid with caustic soda.
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