152 Proceedings of the Royal Irish Academy. 



by direct transformation. Working with the canonical form of UV, let 

 2 = ZX, t = YW, r = - XT, u = ZW, and apply to T, r the linear 

 transformation 



J2Q = ? + it, J2ir = q- it, J2R = r + iu, j2iU = r - iu ; (3) 



the canonical form of Tr is then given by the equations 



To - Q= + ^ + R' + Z7= = 0, 7-0 - 2m {Q-- - T') + 2«, {B' - U'j* = 0. (4) 



Thus, if /x + V - 2m, /z - v = 2/!, we can pass from the one system to the 

 other by replacing m, n, by /t, v. 



If, Ui, JS., E-i be the quantities corresponding to ei, 62, ^, 



j^rw.3 = ,., j^rr^. ^ .„ ^. = ^^ = "-^i^^ ^ 2.,, (5) 



.£'2 = El -V- = 2ei - [m - n)- = - (e, - 2jqi), where »)i = Si - 62 . ej - eg 

 -ff:, = ii'i - f^ = 2ei - («i + nf = -(«,- 2j^), 



and the elliptic parameter for Vr is Pih, where the zeros of F'li are Zi'i, J^a, ^o. 

 If 



Pv, = ^:)M + jiiu + w,) - gi = 2*^ + 



»(i 



pu - Ci 



then 



Pu = Piueto,) = P (w + 0.3 - W2) = P(zi + 2^2). 

 Hence Pu is an even doubly periodic function of u, whose periods are 



2iii = Wl, 2i22 = U,'3 - (Oy, 2Q3 = 2(.)2, 



It has one double pole in its parallelogram of periods at u = 0, and no other 

 pole, and it tends to — as u tends to zero. Hence it is identical with the 

 Weierstrassian jj-function, which has the same periods. 



If Hi, Ei, E3 be the roots of Pu = 0, since pitoi = Cj + Ji^i. 



El = PQ, = c, + J,7, + -Jli = e, + 2j;7i = - E,. 

 E,= - (El +E,) =e,-2j^, = -. E,. 



Ei = PQ3 = C2 + ^3 - e, = - 2ei = - E,. 



_ , ■ . 



Hence (Halphen, t. i, p. 32) Pu = - Piu. 



The equation (6) is the expression in Weierstrass notation of Landen's first 

 transformation. 



* We are using the symbols U, V here to denote coordinates of a connector. This 



not likely to lead to confusion. 



