158 PROFESSOR KELLAND, ON THE 



— 2 = tti m + 2ftg m -i + (tin _2, 



2(l lm -;; + W 4m _3, 



#4m-2 = !• 



These results give 



a Q m 1, «! = - 2, «2 = 3, «3 = — 4, &c. 



asm., = — 2m, «2 m = — 2 + 4m — 2m + 1 = 2m — 1, 



<W, = - (2m - 2) &a « 4m _ 8 = - 2, a 4 ™-a = 1- 



By substitution, therefore, the general expression 



tt r°= t /sin #\ 2 /sin rmx\ % , 



iie / d# . — becomes 



Jo V x ) \ sm rx 1 



— 2 {cos (4m — 2) x — 2 cos (4m - 4) x 



+ 3 cos (4m — 6) x — &c + (2m — 1) cos 2x - m\ 



or _ I **± i c -««-»« _ 2 e -(««-«« + se-w»-«« _ & c . + (2m - 1) e"*" - nl 

 4 « l 



But 1 - 2 + 3 - &c. + (2m - 1) - m = ; 



•. the expression gives ; observing that a = 0, 



+ r-H>(4m- 2 - 2 4m- 4 + 3 4m - 6 - &c. + 2m - 1.2). 

 4 



Now 4m - 2 - 2 (4m - 4) + 3 (4m - 6) - &c. + (2m - 1) . 2 

 m 4m {1 - 2 + 3 - &c. + (2m - 1)} 

 - 2 {I s - 2 2 + 3 2 - &c. + (2m - l) 2 } 

 = 4m 2 — 2m (2m — 1) = 2m. 



