394 Royal Society ; — 



March 31. — Sir Benjamin C. Brodie, Bart., President, in the Chair. 



The following communication was read : — 



" The Higher Theory of Elliptic Integrals, treated from Jacohi's 

 Functions as its Basis." By F. W. Newman, Esq., M.A., Professor 

 of Latin in University College, London. 



The peculiarly beautiful properties of these integrals, as treated by 

 Jacobi and (in his two supplements) by Legcndre, are obtained 

 through so very elaborate and difficult a process, that few students 

 can atford the time to study them. Professor De Morgan, in his 

 ' Integral Calculus,' declines to enter even the Lower Theory, on the 

 ground that the subject requires a detailed treatise. That in some 

 sense it is analogous to trigonometry, which no one would desire to 

 be treated fully in the differential and integral calculus, has been re- 

 cognized by several writers. Legendre, in his second supplement, 

 sixth section, took the first steps toward treating Jacohi's functions 

 (A and 9) on a wholly independent basis, by investigating their pro- 

 jierties from the series which they represent : but after only two 

 })ages of this sort, he aids his research by assuming their relations 

 to elliptic integrals as already established, which shows that he was 

 not seeking for a new basis of argument, but only for new proper- 

 ties. The author of the present paper proposes (for didactic pur- 

 poses) to commence the higher theory from these functions. The 

 first division of his essay is purely algebraic and trigonometrical, not 

 introducing the idea of elliptic integrals at all. Adopting as the defi- 

 nition of the functions A and the two equations 



A(q, x) = 2qi(smx—q^-- sin3a: + (f-^ sin 5x — &c.) \ ^ 



Q{q, x)=.l—2q^'^ cos 2x + 2q^-^ cos 4x—2q^'^ cos 6x + &c. J 

 it demonstrates by direct algebraic methods many properties of great 

 generality, of which we shall here specify — 



1. li Vb stands for -^&i^ and Vc for ^[^'^""1 , which is 



e(q, in) e{q, ^tt) 



shown to yield 5^ + c^ = 1 ; and if, further. A" 0" stand for A(q, x + ^tt), 

 0(2, ^-f^Tr); we get the four equations (equivalent to two only) 



A=+6A"^ = c0=; A''- + 6A-=c0''=; 



0-_ 50«- = cA= ; ©"" - h&-= cA"- ; 



from which it directly follows, that if w is an arc defined by the 



equation \^b tan w= — ^ , we shall have simultaneously Vc sin a*=— ' 



-v/c cos w= ^/h — : Vn — r sin"w)= Vi — The svmbol A(c, id), 



A(w) or A represents V(l— c-sin-'w) in this theory. 



2. It is further shown that 



A»^- A ^= AV^tt) . 00" ; 

 dx ax 



whence is easily obtained 



^OCACcw). Also 0°f^-0^=AXiT).AA». 

 dx dx dx 



3. By direct multiplicatiou of two trigonometrical series, it is 

 found that 



