456 KEroHT— 1882. 



-?J;f-[(ir-?)* (f) -^* (f}>. 



and therefore the condition in order that the difFei'ential equation may he satisfied 

 is that 



must Tanish when taken between the limits and oo of t. 



The integrals (i.), (ii.). (iii.) were derived from known forms of the solution of 



the differential equation (2) by the substitution of ^ for^j + 1 and the use of 

 the formula ''^ 



9. On a Theorem in Elliptic Functions. By J. W. L. Glaishee, M.A., F.B.S. 

 The main object of the paper is to give a proof of the theorem, that if 



nl «! ^^ (n + 2)!^^" (n + 4)!^ ^^^ 

 then the i?-coefficients are such that 



(2n-2)!A2"-'sn=''-'M = i?.«"-i);tsnM + i? (2"-''Ar^sn2<. . . + Ii:l^r,^^'L,--kanu, 



air mr"''' 



(2n - l)lk:'"sn'"u = - H^^"' + lif'khn-'u + Ii['"^^k-snhc . . . + E'lf^^^^ri-u, 



where JR^"^ is independent of u. 



This theorem is proved by Jacobi on pp. 125-127 of the 'Fundamenta Nova;' 

 but the following investigation is somewhat different. 



It is easy to show that if y = e" "^ ^° "■, then y satisfies the differential equation 



1 1 - (1 + A;^).r^ + k\v* J ^^ - I (1 + A^j.r - 2kV | ^ - a=y = 0. 



Let 



y = 1 + Q<').r + Q(2>2, + Q i| + Q(^»j] + &c., 



and substitute this expression for 7/ in the diflerential equation ; we thus obtain, by 

 equating to zero the coefficient of .r", 



Q(«+2^ = ^n-(l + /;«) + a"} QM _ w(m -!)'(«- 2)A2Q'«-« ; 

 whence, since Q"' = a and Q<^' = a*, it follows that 



Q«„) =^«"V +iJ^2»V +Jf^W«6 ...+i?w«2''^ 



the J?-coeflttcieuts being certain functions of k-. 

 Putting arg sn a: = v, we have 



e"" = l+Qm?^%Q<«^+QO'^ + &c. . . (1) 

 1! ^! ol 



and therefore, expandinor e^^ and writing for Q^^\ Q^^- . . . their values, 



= l+«(sntM.i2i3>'-|^\i2«)^ + &c.) 

 o/sn-y wi^sn^y _,,,sn^y . \ 



