PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



26S 



-f + -^ + &c. - ' 





mw - 



y,= 



!\/ — 1 dar' 



1. J 



2jx 2.5.6.7 m^ a; ' 



i.f . V. V 1 



2.5.6.7 TO- ^': 



77 + &C. 



+ &C. 



dx» 



2 ?»■'' 3-8 



&c. 



1 (^'ye .^. 



and y-A y-j^ + — 7=^-^j + B [-^ + _,=.^_^ j 



+ c 



( 



d^i 



d^ 



iJ'K d"^ y 



1 dx^) 



It is scarcely necessary to point out the analogy which exists between the 

 difiPerential equations which determine the value of the transcendentals in this 

 and in the preceding examples. 



14. We proceed now to exhibit a general solution of equations of this kind. 



Ex. 7. 



d^ V 

 y—mx'' — ^ =0 ; w beiViXf any integer. 



The symbolical form of this equation is 



1 - \ + m i-" rf^ 



y = 



l-mx^di 



.0 = 



l-mPx"d^ x"di 



div 



= (l + }nx" di)v = v + mx'' j- (1) 



where v is determined by the equation 



1 



: = D ,• or 



rfi / dh v\ 



Let 



l-mi'x^dix''di 



11 di / dh v\ . 



j-m^ x"- d^x" d^v = or v-m- x"- — , (x" T-i.\ =0 



d^v rf"-i; 



(2) 



dx" 



d^ ( nd^ V 



then 



dx^ d «r" 



P ( ndiv\ nd^z 1 „_lr/"-l^ 

 —i\X r I = X \- -Tztl X ■- 



ir* V dx'^/ dx" 2 rf.r»-l 



1 .1 



.od"-^ 



,_il.l.3...(2n-3):. 



(Part I. Art. 11.) 



