# 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.
- Thus the values of the expressions a 2 /(i - a /V) (Rudolphi, Zeits.
- 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 ...Advertisement
- Let k such transpositions be necessary; then the expression X(kal aa2 N a 3.
- The quantities a l a, a 2 Q ...
- a ll a33 ï¿½ï¿½ï¿½ a32 a33 ï¿½ï¿½ï¿½ a3n an2 an3 ï¿½ï¿½ï¿½ ann Similarly A ik, the cofactor of aik, is shown to be the product of (-) i+k and the determinant obtained by erasing from A the ith row and k th column.
- 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 ...
- f(x) = f =a o xm "- a l + a 2 xm-2 - ...Advertisement
- 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.
- a3 a 2 0 b 2 b1 0 a 3 0 0 b2 This is Euler's method.
- If we write (I +a i x) (I a 2 x) ...
- (I x n x) = I +a l x+ a 2 x 2 -{-...
- from h 2 = a i -a 2 we derive a 2 = h i - h2.Advertisement
- The function Zap 1 a 2 P2 ...an n being as above denoted by a partition of the weight, viz.
- p 1 p 2 ...p n), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as P P3 P2 P4 Pn -2 /, /, /, F i a i 1 a 2 Fi a 1 a 2 ...
- in terms of x 1, x2, x3,ï¿½ï¿½ The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr. Theorem of Reciprocity.-If ï¿½1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ...
- function of separations of (li'12 2 13 3 ...) of specification (si 1 s 22 s 33) Suppose the separations of (11 1 13 2 1 3 3 ...) to involve k different specifications and form the k identities ï¿½1s ï¿½ s Al A 2 A3 ..
- 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,...Advertisement
- 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,...
- where s s a i a 2 a3 = =..
- = ePlal, 1 + a i Q 2+ a 2 0 2 +...
- +bmP"`), a = e?l a' 1 °2 a 2 +..
- an_1 i a n, and in general a n-k a 2 is the symbol for Q k.Advertisement
- 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.Advertisement
- = (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.
- the Hessian (0 a 2 -al)xi + (a 0 a 3 -ala2)xlx2+ (a l a i - a2)x2.Advertisement
- 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.
- Calling the discriminate D, the solution of the quadratic as =o is given by the formula a: = o (a0+a12_x2 (a0x+aix2 If the form a 2 be written as the product of its linear factors p.a., the discriminant takes the form -2(pq) 2.
- 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.
- the complete function may be written ll A2(z) i 2A2 (z/ ' A 2 z 1az2 1.1-a2; and this is the reduced generating function which tells us, by its.
- a little further progress has been made by Cayley, who established the two generating functions for the quintic 1 -a3s 11 -a8.1 a12.Advertisement
- it was noted that Stroh considers Method of Stroh.-In the section on " Symmetric Function," (alai +a 2 a 2 +...
- The general form of perpetuant is (4 K 3 A 2"`) and the generating function 1-z2.1-z3.1-z4 In general when 0 is even and =20, the condition is a l a 2 ...U 24 II(v 1 +a 2)II(a l +a 2 +cr 3)...II(Q 1 +a 2 -}-...
- 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.
- For the case 0=0' =2, the condition is a 1 a 2 T 1 T 2(a 1+ a2) (al +TO (al +T2) = -A2B 1 B 2 -A l A 2 B2 = 0.
- 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).Advertisement
- 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).Advertisement
- 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.
- 3); and this is therefore the coefficient of A 2 a 3 in the expansion of (A+a)5.
- 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.
- u 2 u It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a 2 b 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.
- = b 2 (dx + dy + de l (a 2 - b2) dx (dx+dy+dz) ness where a 2 and b 2 denote the two arbitrary constants.
- 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.
- of the fluid, equal to the weight vertically upward through the movement of a weight P through a distance c will cause the ship to heel through an angle 0 about an axis FF' through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes a 2 -hV/A, b 2 - hV/A, (I) h denoting the vertical height BG between C.G.
- 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).
- Along 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.
- 208.) If 01 denotes the velocity function of the liquid filling the cylinder r = b, and moving bodily with it with velocity Ul, 41 = -U1x, (12) and over the separating surface r =b 4, = I U (+- a2) a2 +b2 (13) 1 Ul b2 - a 2 ...
- 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.
- Thus a 2 vb 2 ?
- In the general motion again of the liquid filling a case, when a = b, 52 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d =- 2+ 2 J, = 2 x22111, d = 2 2`2 (+/'15-Om) (1 yy y n`t dt a +c dt a +c dt a +c) dc2, a2-1-c2 d122 a2 c2 dt ="2) +a2= G2y 71' dt = 121 1 - a 2 -c 2SJ, (19) of which three integrals are e +777 r z y 2= L -?2J2, (20) (a2 + c2) 2 2 121+14 =M+ 2c2(a2-c2)1 ' (21) 121+522hN = + x24 2,2 and then (dt / 2 = (a2 + c 2) 2(° v 2 - 12171) 2 4C4 2 2 - (+ c2)2?(E+77) (?
- In a state of steady motion d4- 121 _S22 Tit °' - fl 4=1G = nt, suppose, S21 -F9,277 = S2co, d4 a2+c2 WI- 1 a2-c2S21' _ 2a 2 SZ dt a2+c2cos' a 2 + c 2 a, 2 a 2 S2 I- a2_c22--a2+C2,0, 1a2 c2)2 (a 2 -c 2) (9a2-c2) ?
- 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.
- A system of confocal ellipsoids is taken y2 (3) a 2 +X b 2 +X c2 + A= I, and a velocity function of the form = x1 P, (4) where 4' is a function of X only, so that 4) is constant over an ellipsoid; and we seek to determine the motion set up, and the form of >G which will satisfy the equation of continuity.
- 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?
- with A' =0 over the surface of the paraboloid; and then' = ZU[y 2 - pJ (x2 + y2) + px ]; (9) =-2U p [1/ (x2 + y2)-x]; (io) 4, = - ZUp log [J(x2+y2)+x] (II) The relative path of a liquid particle is along a stream line 1,L'= 2Uc 2, a constant, (12) = /,2 3, 2 _ (y 2 _ C 2) 2 2 2 2' - C2 2 x 2p(y2 - c2) /' J(x2 +y 2)= py ` 2p(y2_c2)) (13) a C4; while the absolute path of a particle in space will be given by dy_ r - x _ y 2 - c2 dx_ - y - 2py y 2 - c 2 = a 2 e -x 1 46.
- 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)..
- = 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.
- 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).
- Now if the atoms are regarded as points or spherical bodies oscillating about positions of equilibrium, the value of n+3 is precisely six, for we can express the energy of the atom in the form (9 2 v a 2 v a2v E = z(mu 2 +mv 2 +mw 2 +x 2 ax2 + y2ay2-fz2az2), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes.
- - :2 a2A cos e-2 cos (4nnit+e) b2µ b2µ cos f - cos (41rn 2 t +f) 2 2 + a 2 A cos {2a(n l - n 2)t+e} - a 2 A cos {2a(nl-F n2)t+e} a 2 µ cos {21r(n i - n 2)t - f} - a 2 µ cos { 2 ?
- by each animal being about a 2 lb.
- The 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.
- - 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.
- be integers, a l ib i, a 2 /b 2,.
- uredospores in the hemi and a, Fertile cells; at a 2 the brachy forms, and before the passage of a nucleus from formation of teleutospores in the adjoining cell is seen.
- 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.
- ANTHONY WOOD A 2 (1632-1695), English antiquary, was the fourth son of Thomas Wood (1580-1643), B.C.L.
- 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 ?
- where X 2, are any quantities whatever, so that by choosing A 2 b 2 =I, X,X,b3 = I, &c., it can be reduced to any equivalent continued fraction of the form al+ d2+d 3 +d4+ .
- 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.
- 3 bn con - a 2 -a3 -.
- 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.
- Since the plates are very near one another we may use the following equation of the surface as an approximation: y= h1+Ax+Bx 2, h2=h1+Aa+Ba2, whence cot a 1 = - A, cot a 2 =A+2Ba T(cos a i +cos a 2) = +3Ba2), whence we obtain h, = g (cos a, +cos a 2) +6 (2 cot a, - cot a2) h 2 = - T (cos a,+cos a 2) +6 cot a2 - cot al) Let X be the force which must be applied in a horizontal direction to either plate to keep it from approaching the other, then the forces acting on the first plate are T+X in the negative direction, and T sin a t + 2gph 1 2 in the positive direction.
- 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.
- To find the square root of N, we first find some number a whose square is less than N, and subtract a 2 from N.
- 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 ...
- Imagine a curve, real or imaginary, represented by an equation (involving, it may be, imaginary coefficients) between the Cartesian co-ordinates u, u'; then, writing u= x ---iy, u' = x' +iy', the equation determines real values of (x, y), and of (x', y'), corresponding to any given real values of (x', y') and (x, y) respectively; that is, it establishes a real correspondence (not of course a rational one) between the points (x, y) and (x', y'); for example in the imaginary circle u2-{-u'2=(a+bi)2, the correspondence is given by the two equations x '2 - y '2= a 2 - b 2, xy+x'y'=ab.
- =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 ...
- + 2a,a2mlm2+ + a i(n 1+ 25 1+3 K 1)+ a 2(n 2 + 23 2 +3 K 2)+ where a term 2ala2mlm2 indicates tangents which are in the limit the lines drawn to the intersections of the curves P 1 = o, P2 = o each line 2a 1 a 2 times; a term a i (n 1 +25 1 +30 tangents which are in the No.
- 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.
- barque experience will take place on either the J.R. Tolkien, a 2 mast topsail schooner or the Artemis, a 3 mast bark.
- Satin - being fostered in Somerset ywatts 30/06/06 9:05 Satin is a 2 year old ex breeding bitch yellow lab.
- Tusk growth in a 2 year old male wild boar (© Martin Goulding ).
- 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.
- chi-square test for association using a 2 by 2 table.
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- 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.
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- Harvey, gentle rough coated lurcher Barb 26/02/06 2:21 Harvey is a 2 year old rough coated neutured lurcher.
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- microphone amplifier is a 2 transistor direct coupled amplifier.
- 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.
- snowflake curve can be represented in a 2 dimensional plane they have a non-integer dimension.
- 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.
- offering a 2 year manufacturers warrantee you really cant go wrong.
- 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 ...
- a ll a33 Ã¯¿½Ã¯¿½Ã¯¿½ a32 a33 Ã¯¿½Ã¯¿½Ã¯¿½ a3n an2 an3 Ã¯¿½Ã¯¿½Ã¯¿½ ann Similarly A ik, the cofactor of aik, is shown to be the product of (-) i+k and the determinant obtained by erasing from A the ith row and k th column.
- 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.
- 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.
- Taking the same example as before the process leads to the system of equations acx 4 +alx 3 +a2x 2 +a3x =0, aox 3 +a1x 2 +a2x+a 3 = 0, box +bix -1-b2x =0, box' +b i x 2 -{-h 2 x = 0, box + b i x + b:: = 0, whence by elimination the resultant a 0 a 1 a 2 a 3 0 0 a 0 a 1 a 2 a3 bo b 1 b 2 0 0 0 bo b 1 b 2000 bo b 1 b2 which reads by columns as the former determinant reads by rows, and is therefore identical with the former.
- in terms of x 1, x2, x3,Ã¯¿½Ã¯¿½ The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr. Theorem of Reciprocity.-If Ã¯¿½1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ...
- 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 ...
- function of separations of (li'12 2 13 3 ...) of specification (si 1 s 22 s 33) Suppose the separations of (11 1 13 2 1 3 3 ...) to involve k different specifications and form the k identities Ã¯¿½1s Ã¯¿½ s Al A 2 A3 ..
- 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 - ...
- +bmP"`), a = e?l a' 1 Ã‚°2 a 2 +..
- We may then state that D pg is an operation which obliterates one part pq when such part is present, but in the contrary case causes the function to) 171-1-(E7r-1)!7r1 a?2 an!
- 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.
- There Is A Still More General Form Of Seminvariant; We May Have Instead Of 0, 0 Any Collections Of Nonunitary Integers Not Exceeding 0, 0 In Magnitude Respectively, (2 A2 3 A3 ...0 Ae)A(L S 2 G2 3 G3 ...0' Ge') B (12 A2 3 A3 ..0 Ab)A(1 S I 2 G2 3 G3 ...B Ge) B (1 22A23A3 ...0 Ae) A(1822 G2 3 G3 ...0' Ge ') B () 8 (1 8 2 A2 3 A3 ...19'Ã‚°) A(2 G2 3 G3 ...0' ' ') B, Is A Seminvariant; And Since These Forms Are Clearly Enumerated By 1 Z.
- 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.
- Round the cylinder r=a held fixed in the U current the liquid streams past with velocity q' =2U sin 0+m/a; (2) and the loss of head due to this increase of velocity from U to q' is q' 2 -U 2 - (2U sin e to space filled with liquid, and at rest at infinity, the cylinder will experience components of force per unit length (i.) -27rpmV, 27rpmU, due to the vortex motion; 2 dU 2dV (ii.) -71-pa 2 w,, -7rpa dt, due to the kinetic reaction of the liquid; (iii.) o, -7r(a-p)a 2 g, due to gravity, taking Oy vertically upward, and denoting the density of the cylinder by a; so that the equations of motion are 71-0-a 2 - di r = - 7pa2- -- 22rpmV, (4) aa 2 - = -7rpa 2 dV +27rpmV - 7r(cr - p) a2g, (5) 7r or, putting m = a 2 w, so that the vortex velocity is due to an angular velocity w at a radius a, (o+p)dU/dt+2pwV =o, (6) (a+ p) dV /dt - 2 pwU + (v - p)g = o.
- In the general motion again of the liquid filling a case, when a = b, 52 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d =- 2+ 2 J, = 2 x22111, d = 2 2`2 (+/'15-Om) (1 yy y n`t dt a +c dt a +c dt a +c) dc2, a2-1-c2 d122 a2 c2 dt ="2) +a2= G2y 71' dt = 121 1 - a 2 -c 2SJ, (19) of which three integrals are e +777 r z y 2= L -?2J2, (20) (a2 + c2) 2 2 121+14 =M+ 2c2(a2-c2)1 ' (21) 121+522hN = + x24 2,2 and then (dt / 2 = (a2 + c 2) 2(Ã‚° v 2 - 12171) 2 4C4 2 2 - (+ c2)2?(E+77) (?
- In a state of steady motion d4- 121 _S22 Tit Ã‚°' - fl 4=1G = nt, suppose, S21 -F9,277 = S2co, d4 a2+c2 WI- 1 a2-c2S21' _ 2a 2 SZ dt a2+c2cos' a 2 + c 2 a, 2 a 2 S2 I- a2_c22--a2+C2,0, 1a2 c2)2 (a 2 -c 2) (9a2-c2) ?
- (12) Along the stream line xBAPJ, t ' =0, u=ae-" c bl, n; (13) and over the jet surface JPA, where the skin velocity is Q, - q = - Q, u = ae rs Q /m = ae rs lc, (14) ds denoting the arc AP by s, starting at u = a; a ' ch nS2=cos nB= -a' u u - - a b' (15) a l a - b l u - a' a-a' u-b' co > u = ae'" S " c > a, and this gives the intrinsic equation of the jet, and of curvature ds '&1) _ i dw i dw dS2 P= - dO = Q a0 - Q as2 = Q c u-b d (u -a.u -a') _ ?
- 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 velocity perpendicular to the axis of any point on the curve at a fixed distance x from 0 is dy_ (I ]) at A A The acceleration perpendicular to the axis is d2y = 2 2 dt 2 - A 2 sin A (x - Ut) The maximum pressure excess is the amplitude of ("6= Eu /U _ (E/U)dy/dt.
- 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.
- When boiled with caustic soda it isomerizes to a mixture of the A 2.4 and A 2 'Ã‚° dihydrophthalic acids.
- 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.
- Finally after traversing the analyser the sum of the two resolved components is a cos (l i t - a) cos 41 3) cos T +a sin (p - a) sin (>G - a) cos (T - p), of which the intensity is {a cos (1k - a) cos (,f,-0) sin sin (-0) cos a 2 sin 2 (tP - a) sin' (-0) sin 2 a 2 cos 2 (13 - a) - a 2 sin 24 - a) sin 2(>!
- THE OPERATION The conventional operation involves a 2 - 4 inch long cut in the right lower quadrant of the abdomen.
- Left end adjoins No. 46 whilst right end has a 2 light both floors and framing forming quatrefoil in gable.
- Each GPM flowing through the tee results in a 2 ft/sec velocity which performs the self-cleaning action.
- Fractal dimension: Although shapes such as Koch 's snowflake curve can be represented in a 2 dimensional plane they have a non-integer dimension.
- This in conjunction with a 2 speed bottom bracket gear made the six speed Sunbeams possible.
- The 2 nd team managed a 2 nd place finish with 15 points, whilst the women stormed their league with 6 consecutive victories.
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- Sew with a 2 ½ millimeter zig zag stitch for the best results.
- Sears offers a one percent discount for those who purchase between $25,000 and $75,000 in gift cards or a 2 percent discount for those purchasing between $75,000 and $149,999.
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- Ideally, the vows should be between 250 and 500 words, which is equivalent to a 2 to 4 minute speech: shorter vows may seem rushed and insincere, and longer vows will seem too drawn out and overdone.
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- It's a 2 on 2 matchup in Beach Volleyball where you bump, set, and spike your way through Arcade, Exhibition, Tournament, and training modes.
- Like the original, Lumines 2 gives you a huge grid on which to play, with a series of slowly dropping blocks, consisting of a 2 x 2 square of multicolored boxes.
- If you do the later, you take control of the fighting and it's possible to have a 2 on 3, 1 on 4, or 3 on 3…depending on if you sent your guys into help.
- Text Twist has a 2 MB downloadable version with a few perks.
- If a player rolls a 2 and a 3, for example, they must put the checker on the bar on either point 2 or 3 of their opponent's inner quadrant.
- In addition to an amazing 470MB of internal memory with Memory Stick Duo Pro expansion, the Sony Ericsson W900 also sports a 2 megapixel digital camera, UMTS high speed data, Bluetooth, and a suite of 3D mobile games.
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- The Prissy is a sexy strappy sandal with a 2 3/4 inch heel, leather upper, lightly padded foot bed and elasticized ankle strap.
- X-strap, is fully lined and cushioned with a 2 1/2-inch contoured heel and a non-slip suede sole.
- T-strap boasts a 2 1/2-inch contoured Latina heel, fully flexible forepart, suede sole, lining and socklining.
- There is also a 2 1/2 inch version of this shoe.
- A quarter-sleeve space on the upper arm, the back of a shoulder blade (at least a 2 inch x 3 inch area), or the middle back are good choices.
- Cut out a 2 inch by 2 inch yellow square.
- This amount includes what was formerly the employee portion, the part that the employer paid and potentially a 2 percent administrative fee.
- Choose a fishnet bodysuit with a 2 to 3 inch wide weave.
- The almighty Stax label (Issac Hayes, Otis Reddding, The Staple Sisters, and more) is celebrating its 50th anniversary this year, and to kick of the festivities, they have released a 2 disc set of some of their greatest hits.
- The laws of branch formed by similar budding from 1; budding in hydroids a 2 -d 2 from 2, and so forth.