Xumerical Calculation of tlie Jacobian Ellipsoids. 



7. As li tends to unity, the foregoing equations become 

 gradually inconvenient. In tliis case, the ^-functions ^vith 

 modulus r may be transformed into those with modulus r^ = — r-^, 

 according to the formuke 





(29) 



where a- denotes that the modulus of the function is z,. By this 

 transformation, h goes over to /^j, wliere 



lognit// ■ lognat//i = tz'. 



"(30) 



It must be borne in mind that the values of A,, with which we 

 shall be concerned hereafter, are exceedingly small, being less than 

 0-0009920 which corresponds to the extreme value 0-24: of h 

 referred to al:)ove. 



By the modular transformation, (14) becomes 



(31) 

 In this case, an equation analogous to (17) cannot be obtained, for 

 the argument z^v increases without limit as //•, approaches zero. 

 We may, however, proceed thus: — 



1 



Writing 



= —'IrziT.v = ^^Icgnat- 



h: 



m 



and substituting for the functions in (31) their respective 

 /ii-series, this equation may be put into the form 



(ÄU+BV)z, = CX+Y, 



where U = cosh z, + //,-cosh 3^^ + h.'^cosh 5z, + , 



V =l + 2Ji, cosh 2z, + 2h,' cosh éz, + , 



X = h, sinb 2z, + 1h ,' sinh 4.r, + , 



Y = D ahûxz, + Di/i, sinh 3,?, + DJ/, ^ sinh 5^ + 



(33) 



(34) 



