TRANSACTION^; OF SECTION A. 



549 



I 



sin u du= — cos u, 



tan u du = — log cos u, 



sec udu = log tan (^ M + J^ tt), 



1 



COS u du = sin m, 

 cot u du = log sin m, 

 cosec u du = log tan ^ u. 



four of the integrals being thus expressed as logarithmic functions. In Elliptic 

 functions the complete system of in tejirals may be expressed in the logarithmic 

 lorm — viz., we have : — 



sn u 

 dn ;< 

 sn u 



r CUM 



J SUM 



r dn tt 

 J en tt 



J cntt 



du = 



du = 

 ■du=: 

 du = 

 du = 

 du= — 



- log 

 loj; 



on ^udn^ u 



sn ^ u 

 sn 1^ M dn ^ M 



en A M ' 



J sn i ticn^u 

 " dn ^ M ' 



_ log ca(^u + iK)dn(iu + ^K) 



sn(^u + ^K) 



CUM 



sn u du = — 

 en u du - 

 dnudu = 



cnu , --, 

 , du = ^ 



1^ 



k' 



k' 

 1 

 k 



i 

 k 



loff sn(g'* + ^-g)dn(^» + ^JC) 



Ir^n- ^T^(iu + i K)cn(iu + ^K ) 

 " dn(iM + ^i:) ' 



lo„ en (iu + iK')dn(^ 71 + ^ JK') 

 sn {^u + ^iK') * 



lo^ sn (^ M + 1 JK') dn (^ » + ^ »^^ ) 

 cn'{hu + ^iK') ' ' 

 , sn (\ u + ItXPcn (^ u + ^ tXQ 

 ^ dn (i « + ^ iK') 



1 io„ cu (i M + i .g^- j- JK') dn r^ M + ^ -g+ ^ JK') 

 ''du=- J_ Incr S" q ^< + ^ ^+ i eXQ dn (^ w + ^ g+ ^ iK') 



sn (in + ^K+^ iK') cn{^u + ^K+^ iK'). 

 ~daQu + iK+liK') 



I 

 1 

 I 

 f 

 I 

 I 

 I 



The system of integrals may be written also as follows : 



dmt 



sn ?« 



du = 



— loo' 



k' ' " 



1 t^w= -ioglnjf±£^, 



SUM SUM ' 



fdn u 



J SUM 



1 



du = 



lof 



1 — cnM 



sn II ' 



cnu , 



du = 



sn le 



fdn u 



cnu 



du = 



1 1 — dn M 



log , 



snu 



-log ir:l°«, 



CUM 



I 



f 



cu M A' " en M 



