VARIABLE 75 tions to elliptic functions of a complex variable. Calling K(z) = (z - a)(z - b)(z - c)(z - d), then Atf(7) will be ex pressed as a unique function by means of a two-leaved Riemann s surface. Its branches take the same values at each of the points a, b, c, d. Let the leaves be pierced through and joined transversely along the lines a to b and c to d (fig. 6). Assume that in the upper leaf /-K(z) at the origin is + It and in the under T leaf is - R. Every path which, when projected on the upper leaf, goes an odd number of times round an odd number of branch points leads from one leaf into the other; on the other hand, a path which goes an even num ber of times round an odd number of branch -points, which oes or round in value accrue to Fig. 6. an even number of branch-points, ends in the leaf in which it originally began. Now let us investigate what changes dz f and to I JE(z)dz when z de- /n(z) J v scribes part of a vanishing circle round the branch point a. First, let z = a + pe ie , then dz = ipe ie dO, R(z) = pe io (a - b + pe ie )(a - c + pe ie )(a - d + pe ie ) = pe w S. Hence . J 3 = iplgWdO f S. But, when p is evanescent, lim / S = [(a,-b)(a-c)(a - d)] = T, and is therefore neither infinitely great nor evanes cent ; integrating therefore between 6 Q and V we find _ i ,(?! ___ ^ r<ji limv//>. - J e is 2 dO and lim V 7 J et i8 d0. Since these 1 J 1 J 60 vanish when p does, and the other factors are not infinite, we see that, when I -7==; and I */fi(z)dz&TQ taken along J v R(z} J paths surrounding indefinitely closely a single branch point, both these integrals vanish. If now we confine our attention to the integral of the first kind, let z go from the origin in the upper leaf along a straight line to near the point a ; then the integral - - , /R(z) putting y mx in z = may be calculated as for a real variable, and it increases [ a dz by I / - ~T- Next let z go round a in a vanishing circle, the integral gains nothing ; then let z return to the origin along the same line, it proceeds now in the lower leaf ; thus both /S(z) and cfehave signs opposite, but values equal to, what they had on the outward path. Thus on the whole by travelling this elementary contour (lacet, Schlei/e) from O to near a, round a, and back to O, the value of the in- f a dz tegral is increased by A = 2 I j= In like manner we / N/A (*) have for the elementary contours or loops round b, c, d f> ( re ( the three increments B 2 / - C = 2 = -, ./ov^) JoAff(*) and 19. Now, if the loop round a be described twice in suc cession, the integral changes by A - A and so on. But if, after a is once described, b is described once, the change is A - B. If then z go straight to any arbitrary point for which the value of the rectilinear integral is iv, the value along the entire path, or along any path recon cilable with it, is A - B + w. The two loops round a and b may be taken as often as we please before going by the straight course from O to z ; thus A - B is a period of the integral, which is accordingly an infinitely many-valued function of z. As there are three independent differences of any four quantities, it would seem as if there should be three periods ; but we can show that A - D arises from a combination of A - B and A - C. In fact, describe round the origin a circle to include all the four points ; then let 3 go straight from the origin to a point p on this circle, travel round it positively, and return from p to the origin. The integral has along this path the same value as along the four loops in succession, since between the two paths there is not any singular point and both have the same origin and goal. Thus the value of the integral along this path is A - B + C - D. But, when z goes round the circle from p, the radical changes sign four times. If loops be described from p round a, b, e, d, they are recon cilable with the circumference of the circle. Hence the radical has the same sign when it returns to p as that it set out with ; accordingly, since along Op and j)O the radical has the same sign, the integrals I(0p) and I(j;O) cancel, and the integral along the four loops in succession has the same value as along the circle alone. To find this value, put z = pe i9 and it becomes . But, as PJ P/ P/ the radical is unique along the circumference of the circle, the value of this integral is the same whatever be the value of p, provided it be large enough to include all the singular points. Hence we can let p increase indefinitely and then this has the limiting value zero. Consequently A - B + C - D = 0, or A - D = A - C - (A - B) ; thus the third period is composed from the other two. Hence the periods reduce to A - B and A - C. It can thus be shown that all values of the integral for all possible paths of inte gration between O and z are included in the two formuhe m(A B} + n(A C) + iv&udm(A B] + n(A C) + A - w. 20. If a new variable z be introduced, homographically related to z by the equation a + /3z + yz + Szz = 0, it trans forms the integral!- ,-" - into (a8-/3y)l . ". r . The J v Ji(~) ./ -K-i(z ) coefficients of R^z } can be determined so that it may differ only by a constant multiplier from (1 - z -}( - k-z -}. As k is given by a reciprocal quadratic equation, it is always possible to choose such a value that its modulus may be less than 1. In other words, it is possible by a homographic transformation to convert the quadrilateral a, b, c, d into a parallelogram, the new origin being at the intersection of its diagonals (fig. 7). This leads to Jacobi s form of the integral when k is supposed real. The inte- _of 1 <** J<y (i-^xi-^vy js=2i llk - =4^= c-* f ~ l grals along the loops are here dz Hence yi-<7 = 2 A - B is the integral taken consecutively along the loops of a, b ; but for these paths we may substitute the following : proceed straight from O to a near a, then round a back to a, then straight from a to /3 near b, next round b back to /?, then again straight to a, and straight back to O. The integral takes the same value by this path as by the other, since there is no singular point
between them and both paths have the same origin and