ON ELLIPTIC ANU H VPERELLIPTIC FUNCTIONS. 323 



It is easily seen that if we change ti, u into zt+4K, w' + 4K', or into 

 tt+4L, m' + 4L', all these series remain unchanged. Hence they are doubly 

 Iieriodie. Moreover their ratios are quadruply periodic ; for after removing 

 the common factor ei't^'+ru-^ gjj ^]^q exponents of e in numerator and deno- 

 minator are linear in ii'a. Hence it is easy to prove that if u, u are changed 

 into M+4A, M + 4A' when 



, ttL f it TrLt 



4r(KL' -K'L)' 4/(XL' - K'L/ 



or if a', u are changed into m+4B, m' + 4B' when we have 



B= '^'L-, B = - '^^^ 



"4;-(KL'-K'L)' 4y'(KL'-K'L)' 



the ratios of these functions remain unchanged. 



If we suppose «, v' to be augmented by the semiperiods, the quantities P, Q, 

 &c. sometimes remain unchanged, sometimes change their sign. The resulting 

 values are expressed by Gopel in a Table, where the first line gives us the 

 increments of the argument, the remaining lines the resulting signs, thus : — 



2K, :2L, :?Iv + 2L. 



- - + 



+ 



and so on for the remaining fourteen series (Gopel, p. 282). 



When we suppose «'«' to be augmented by the quarterperiods, P, Q, &c. 

 are changed into other functions of the series, as is expressed in a Table, 

 where the first line, as before, gives us the increments of the arguments, the 

 remaining lines the quantities into which P, Q, &c. are changed, thus : — - 



A B A + B K L K + L 



P' P" P" ;a ill 8 



Q 



a' a" Q'" »p — «s -R, 



and so on for the remaining fourteen series (Gopel, p. 283). 



Gopel next gives a Table of the values of «, u', which cause P, Q, tfec. to vanish ; 

 thus U vanishes for 0, B, A + L,K+L, B + L, A+K + L; Pfor K, L, A + L, 

 B4-K,A-fK + L,B-}-K+L; all the ftmctions multiplied by (;) vanish for m=0, 

 u' = 0. I may remark that the vanishing of functions d has been treated in 

 detail by Eiemann, in the 6oth volume of Crelle's Journal. We shall refer 

 to the three Tables described in this section as Giipel's first, second, and third 

 Tables. 



Section 11. — Gopel next investigates the algebraical relations between the 

 functions P, Q, &c . . . . In doing so he makes use of the following notation. 

 If in the functions P'", Q,'", 11'", S'", 2/-, 2r' are written instead of r, r', the 

 four results are denoted by T, U, V, W. When in these functions v and n' 

 vanish, the results arc denoted by f, u, v, iv ; consequently u is used in two 

 different senses in this paper. I shaU endeavour to guard against any con- 

 fusion arising from this. When u, v! arc put equal to zero in the functions 

 P, Q, R, S, P', P", &c., the results arc denoted by m, h, p, a, -ar', •nr", &c. 



Then by direct multiplication the following formulae are arrived at without 

 difficulty : — 



P==<T-mU-i'V + u;W, 



V 2 



