206 



Mr. J. Griffiths. Doubly Periodic Elliptic [Mar. 12, 



III. " Abstract of some Results with respect to Doubly Periodic 

 Elliptic Functions of the Second and Third Kinds." By 

 John Griffiths, M.A. Communicated by Prof. Stokes, 

 Sec. R.S. Received March 5, 1885. 



Elliptic Functions of the Second Kind. 



c 9 de _p de f ff de 



L If M= ) 7^^kHitf e , K Jo-Zl^Fri^' K= J ^i^PSrI^' 



Jc 2 -\-k' 2 = l, e=amu, sin 0=sin anm, cos 0=cos anm, 



— Jc 2 sin 2 e— Aanm, it is known that — 



sin am(« + 4K + 4?'K') = sin am?<, 



cos am(^+4K + 4{K')=cos anm, 



Aam(w + 4K + 4^K')= Aamit. 



Similar results are true when u is replaced by the second elliptic 



f 



integral a= I VI — 7s 2 sin 2 0e20, cmo 1 a corresponding change is made in 

 Jo 



Jacobi's notation. 



In fact, putting 0=ama, sin 0=sin ama, cos #=cos ama, 

 -s/1 — & 2 sin 2 0= A&ma, we have the following theorems, viz. : — 



sin am{ a + 4E + 4^(K' — E') } =sin ama, 



cos am{a+4E + 4i(K'— E')}=cos ama, 



Aam{a + 4E + 4i(K / — E')} = Aama, 



if E=P yi-^sin 2 ^, E' = T ^l-y^sm 2 ^, ^+J^=1, and 



Jo Jo 

 K, K' are the same quantities as above. 



The fundamental equations satisfied by the new functions sin ama, 

 cos ama, Aama are — 



(1 .) sin am{ a + b — W sin ama sin am&/(a, b) } =/(a, b) , 



(2.) cos am{a + b— h 2 sin ama sin amfc/(a, &)}=0(a, &), 



(3.) Aam{a-r-fr — h 2 sin ama sin am&/(a, b)} = f(a, b), 



