^94 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



1 nf / 4r + l / — T • 4r + l \ j—z ) 



= — ??«" -^ { cos — ~ — nnr + v — 1 sin nnv \ e™-'= ^— ^ ( 



+ j cos ( 2 /* — — J n TT + \/ — 1 sin ( 2 / — —J « tt 



g—mx V— 1 



= -^- ?«" i COS ( 2 r + -y . « TT + »« a; j + \/ — 1 sin ( 2 r + -^ « tt + ;w ,r j 

 + cos (2/ — -^"TT — ma;J + \/ — 1 sin / 2 / — -^«7r — m x| i 



= ;/<" \ cos r + r' nnr , cos . (r — i-' + — n tt + m x\ 



+ \/ — 1 sin r + »' » TT cos ( r — / + — . n TT + w? a; J v 



= y/^" I cosr + /«7r+ \/ — 1 sinr + ?-' . «7r i cos W~''' + -o ■ ■"'^ + >'«-^ ) 



,„ ^ ^ , d" sin J7? a: 



23. Tojind 



dx" 



f/''sinma:=-p-^jS=^|(\/-l)V'^^-' - (-\/-l)" e-»*V-i| 



so that the quantity under the bracket differs from that in the expression, for 



d" cosmx 1.1./ .11 



— J-;, only in having (n-l) m the place of n. 



TT d"smma; „ ( — — - ^ / — ^ . , :, ) 



Hence 7~^ =m" Acosr + r'.n — l'rr+ V — I smr + r . n — 1 ir \ 



dx" 



X cos 



I r—r' + -^ .n — lTT + ffix] 



Cor. 





rf" + ^ . sinrnx 

 dx" + ^ 



. = 



m 



d" 



■ cos »M a; 

 dx" 



Ex. 



If 



m = l , n = 



1 

 2 



5 



we 



get 





—. = I cos r 4- 



7^ 



J. a/ 



— 1 sin i 



— = \ COS r + r . —- + V — i- sinr + r^ --\ cos (/ — / + -^-; + 

 c?a;2 I 2 ^j \ ^ z 



Now we may give to r and r' any integral positive values we please t 



r}=0 gives cos ( — + x\ 



Let r'=0,J /Stt 



ir = l ... a/^ cos ( ^— + r j 



