PROFESSOR KELLANT) ON GENERAL DIFFERENTIATION. 2§7 



Again, let ^== ^^ + r^i^ +^'- 



then ^^ (V^^ J = - Y7^ - &c. 



1 



a 3 J 3 



a^zp^ 



and y=A(yi--^y.)- 



By solving the equations for y, and ^/^ we obtain finally 



The equations fi'om which i/^ and j/,, are determined differ only in the term 

 which does not contain y ; and it will be seen hereafter that similar equations 

 serve to give the solution of the other differential equations of this class, when n 

 is an integer. If a\/^^=m, these equations are 



dx \2x TO^a-V-^i 1x 



dx \2x in'x')^' 



12. Otheewise. The following method of solving this equation has the ad- 

 vantage of not appearing to take for granted the form in which y is expressed 

 in terms of x. 



I d^ V ■ 



y-av-\x — 1=0 gives y = 



dx^ 1-av -Ixd 



1 + a'^ —Ixd 



\ + a^xd^xd^ 







Now T — i is the solution of the equation 



l + a^xd*x(P 



v + a? xd'' xd^v-Q, or of 



V „dv \ „ 



a- dx 2 



dx \lx a-x-) 

 which is the equation for determining y^ given above. 



VOL. XVI. PART III. 3 T 



