j{^4 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



= in" I ( cos — ^^- — nir + 'J~—i sin — — - — « tt j c" ' ^— i j 



+ [cos ( 2 ^^ - ^ j M TT + \/ - 1 sin ^ 2 r" - — J M TT 



g— mar V~l 



= -g- >»" I COS f 2 r + -^ . « TT + »< « j + i/ — 1 sin ( 2 r + -j- « tt + «? ,r | 



+ cos (2/ — -^"TT— war) + \/— Isin f 2?-' — -:y WTT— »«3| I 



= «(" ] cos r + r' « TT , cos . (»• — / + -—« tt + ?« a- 1 



+ 



\/ — lsin>- + /«7rcos (>■ — / + -J- • nTT + ma-j i 

 ///" lcos7+7'«7r+ s/^ sin 7+7-' . «7r i cos (»•-(-' + ^ . ii'K + mx\ 



„ , r/" sin jrea: 



23. To find. 



rl" •immx= 



dx" 

 m" 



/(-/_!)" ,."■ •'■ n'-' _ ( _ -v/ _ 1)" e-'" ^ ^-1 I 



2V^t r 



.so that the quantity under the bracket diflfers from that in tlie expression, tor 



(/"cosmx , . , . , . ,T , n 



— j^, — only in having {n—1) m the place ot ti. 



jj d" sin tnx , ( , =- / — =- . ■. = ) 



Hence ; =»«".< cos?- + /.?j — l7r+ v —Ismr + r .n—lir} 



dx" ( ) 



X cos (r—r' + -^.n — l7r + »ix\ 



„ d"'^^ .sinrnz d''eosmz 



Cor. --— ; — = m . 



dx^ + ^ dig' 



Ex. If »?=1, n- — , we get 



_ , .,js »■+/ . J-+v' — 1 sinr + /— I cos ( ''— '' + -^ -^ + -^ ) 

 dx^ I 2 2J \ 2 2 / 



Now we may give to r and r' any integral positive values we please : 



5/=0 gives cos I -j — V x\ 

 (r = l ... v/-l cos f ■-^- + .rj 



