76 The Rev. B. Bronwin on Jacobi's Elliptic Functions. 



theory. As co has n for a denominator, we must have u = n(a, 

 or some odd integer multiple of this quantity. But s a [n w) 

 = 1 for the first form of w only. I expect Mr. Cayley will 

 object to this, and think that some other value of n may be 

 found to satisfy the required condition. Let then ^j/iw be 

 such a value, and let us determine it so as to satisfy the re- 

 quired condition for the third and fourth forms of co if pos- 

 sible, in order to oblige that gentleman. 

 By pages 32 and 34 of Jacobi, 



sa{pnw)) = sa {pmK +p m' i K'), 



{s a {p m K) c a {p fn' i K') A.a {p m' i K') "\ 

 + s a{prn! i K') ca{pmK.) i^.a{pmK) j 

 1 —k'^fi^aiptnK.) s'^ a (p m' i K') 

 f sa(j7wK) A.a(;)m'K')+ ^ 



_\isa[pm^K.')ca{pm'K!)ca{pm'K) ^.a{pm'K)j' 



" c^a{p m' K') -Vk'^s^a{pm K) s^ a {p m! K') * ^"^'^ 



The modulus of cw, and therefore of « K', is k; but that of K' 

 in («.) is k', its own modulus. 



The imaginary quantity i must vanish from (a.). Let then 



2 r 

 sa{pm' K') = 0, or p7n' = 2 r, p = — y-, 



r as well as m and m' being any integers, positive or negative. 

 The result is that 



{a.) = sa{p7nK) —sal — — K ). 



If — — be a fraction, this cannot answer; if it be an integer, 



it is an even one, and gives sa (ptnK) = 0, which does not 



answer ; for m is even, m' odd, or both are odd. 



2 /■ + 1 

 Next makeca(2; w' K')=0, or pm' = 2r-{- 1, p=: j — , 



The result in this case is 



^""•^ "^ /csa{pmKy 



which must be greater than unity, and therefore cannot an- 

 swer. 



2 r + 1 

 Make c« (» wK)=0, or 2;;» = 2r-f 1, » = , 



the result is 



^'^•^ ^ A,a{pm'K'y 

 which cannot be unity for 7)i even, m! odd, or for both odd. 

 The third and fourth forms of m therefore never can give 



