550 KEPORT — 1881. 



du= - y, log , 



en M k en u 



I sn M rf?/ = J log (dn m — A; en u), 



lcnMdw= ^ log(dnu-iX;8nM), 



dn M rfw = i log (en u — i sn m), 



fcnu J 1 1 1— A;8nM 



3 — dll'=-- log — 3 , 

 dn M k dn ii 



fan M , t 1 A en M + lA;' 



3 — du= -J— log — ^ , 

 dnM A;A' " dnw 



fl J t , en M — ik' sn m 



5^^^"= F ^°g dn u ' 



Only four of these expressions are imaginary, and by reducing them to real 

 forms, we obtain the formulae : 



J 1 , A; sn M 



en u aw = — arc tan ^j , 



k duM 



J , , sn »« 



dn M OM = arc tan , 



en ti 



' fsn M , 1 , k' 



- — du = ,-— arc tan , 



J dn M kk k en 



— > 



u 



1 J 1 . A' sn w 



— du = -~ arc tan , 



dn u k' en M 



Formuke of Reduction for sn" m dti, &c. We have 



^sn" M = M (n - 1) sn"-= m - wVl + ¥) sn" u + n(n+l)k'^ sn«+2 m . . . (1) 

 dir 



and by means of this formula the integral of sn" u may be reduced to depend upon 

 the integrals of sn m, sn^ u,- ^ — — according as n is positive and uneven, posi- 



SIl U SU It 



tive and even, negative and uneven, negative and even. 



The integral of sn'^w involves the Zeta function; and the integral of — r— 



may be deduced from that of sn" u by means of the formula : 



— - log sn z< = A;- sn^ M — _: (2). 



dw sn'-M 



Corresponding to (1) and (2) there are eleven other pairs of formulae which 

 involve the other eleven functions in place of sn u, and difi'er from one another 

 only in the A;-coefEcients, It can thus be shown that the integrals of the w"^ 

 powers of the twelve functions are all finitely expressible in terms of elliptic func- 

 tions if n is uneven, and in terms of elliptic functions, and of the Zeta function, if n 

 is even ; and that the twelve formula of reduction are similar in form and such 

 that from any one of them the other eleven may be deduced.* 



' The paper will be printed in extenso in the 3Iessenjer of Mathematics. The first 

 portion appears in the number for October, 1881. 



