IN THE DIFFERENTIAL CALCULUS. 53 
14. All equations of the form y—m 2’ Stax can be converted into equations 
dite 

ae 
Ga 
a n D+ 
For (ie cee "y=X 
Let y=|—D.», then 
j= cance /-D+nv=X 
n D+ r—n ——— 1 
or v—m(—1) ae re ea =p) Xx 
—(r—n) a d * r—n —..-1 - 
or —m(—1) x (a x v= (/—D) 
Ot. it beet aa — iz 
z—m(—1)~ (r—n) y2r— sae (=D ox 
wherefore the above equation is reduced to an ordinary differential equation 
whatever be 2, provided 7 is an integer. 
As an example, let us take the simplest case, of which the solution will be 
found at pp. 257, 258 of my Memoir on General Differentiation, Part III 


dy 
Ex. 1. y—aV/—12 aa 0 
Suppose y=/(—-D2, 
then edara ve/—DET Gy vt) = OVz 
3 dz 
or 2—az dan ON% where z=v/z; 
ee 
pags avef A C ae gv) 
ae a. x 
ech 2 
Cae eat oe Ere 
e@ te aVx , 
and y=|—D Ja {-2-£ cee pee a.) 
a 


ae Mages 
In fact, it will be 
2 
fron gH Sfp 
— é ee 
° Jp 
The constants A and C are not ath of them arbitrary 
A 
seen that C= ip: so that the final form of equation (1.) 

— 9 
— 2 
Re (jo S evel 
