whence we have 



TRANSACTIONS OF SECTION A. 457 



o/sn'y w.iSn^y -r^f7^sn'y „ \ 



+ &C., 



-,(3,sn''y ^,= ,sn''y „ 

 V = SUV + -Ri -ITT + -Si -KT + ^^-i 



•■,2 2 



and generally 



&c. &c. 

 v" sn"« w^j.91 sn"+2y -^,„j.a^ sn"+-'y 



nl n] ^ ^" (n + 2)! ■ '"" (h + 4)! ^ 

 so that the H's are the coefficients occurring in the expansions of v, v^, y' ... in 

 ascending powers of snv. 

 Now from (1) we have 



^(ea.- + e-«t.) = 1 + QC2^^ + QC4)!^ + &c. ; 



replace a by ^, and let the function operated upon he sn u in the first equation, 

 and sn-M in the second ; then since 



e (l>Ui)=cj>(u-T v), 

 we obtain the formulae 



I \ sn u sn 2j 



i jsD(?<+y)-sn(M-y) . =Q'("sm6.sny+Q'<'isuM.-gj^+Q'''^snM. gj- + &c. 



i ] sn\u +v)+ sn- (u -v)\ = sn-?t + Q'^^hn"u . !^ + Q'W)sn% .— + &c., 

 ' I 2! 4! 



where Q'""-" and Q'^^"' denote respectively 



T, (in-l)d (in-l)d^ (2n-l)d^ ^ j.12,,-1) d^"'^ 



du * dw^ '- dw- ' ' ^""' du^"~^ 

 Therefore by integration 



and If '-'"_^ 4. 7? ^2")<^' , 7? (2")<^® , T?(2«) '^'' 



J 



« } sn (it + 1') - sn (u- v) l <?m = Q"("A; sn ?t sn j; + Q"'^^^k sn m -;^ + &c. 







uTu 



^ri /c^ I sn= (m + y) + sn' (m- y) -2 sn *m j. du du 



I (2) (2) ) sn "r 1 w <^' 1 sn ^y 



= I Q" F sn -u-S^ ^ + Q" it- sn -u-S^ [ ®-^" + &c. 



where Q"(2«-') and Q"'*™) denote respectively the operators 



J, (2«-n , T> (2n-l) t^'^ , „ (2«-l^ '^■' _^ p(2«-l) <P"~' 



and iJ. ""^ + i2,'^»'^ + i?/^">-^ . . . + <";^,, 



du^ "^ (?M^ *» rfw"'-* 



and i?g denotes the value of 



(2ll) Ji C2«) A* (2n^-72ll-! 



-R. 3^„A;2 sn % + Jf?6 ^„A;' sn ^^j* . . . + J? ^3—- jA-' sn »w 

 flit* ° du' j„ du'"' 



when M is put equal to zero. 



