41 



in the form d "' <p Qx, y) ~ o, such equations will servefor determining 

 certain of the values of Cj , Cg &c. 



It will be generally found more convenient in reference to the 

 converse of theorems foK finding fluxions per saltiim to use the 

 fluxional equation. 



dm $ (x, 1/) 



1 . 2 . .m ~ ^ 



which may be denoted by 



d"' <p (x, y) = o, the denominator being understood by the 

 line under d and by the order of the fluxion, so that 



dr <p {x, ij)=o 

 may with respect to the value of a; = o and the denominator be cal- 

 led a particular divided fluxional equation, of which the integral 

 is 



(p (x, y) = Ci + Cz X + Cs x^ c„ x"~^ 



This mode of proceeding is applicable to the summation of many 

 series, to finding generating functions, to the integration of equa- 

 tions of finite differences and of many fluxional equations, more 

 particularly those equations called linear equations. 

 To proceed with the equation (e) 

 It fluxion gives 



d^ u -j- dudr = m^ Kc sin (Ck— tr) d i>^ (I) 



1. Taking the n — 3 particular fluxion when y=o and d y is con- 

 stant, 71 being even and not less than 4. 



n— 4 



d" u + d^-^udv"^ = m" ( — 1) ~ Kc"-" cos ardf" (2) 



2. Taking the n — 3 particular fluxion when t—o and dt u con- 

 voL. xm. H 



