284 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
(p'd+q di+9) (d+adi+b)x=(pd+qd?+r)(d+ad+v)z 


/ ad x / / / d? x av if 
or (p—P’) Fda da dae Mae Oar eo 
t d t? 
, dx U / d? x 
—p b—ga-r) 7 +qbtra'—g b—ra) if 
+rb'—r b6)x=0 
which coincides with the general form Cor. 2, Ex. 3, Class 1. 
Section III. Parriat DIFFERENTIAL EQUATIONS. 
21. In order to effect the solution of partial differential equations in which 
the operation with respect to x is totally independent of the operation with 
respect to y, we must distinguish the operation of differentiation in the two cases 
by different symbols. Let d stand for = 0 for i : then in solving the equation 
with respect to d, we may treat 0 as a constant, and vice versd. 
Bx. 1. Bs #2 Wo, 
dx? dy* 
Write this equation (d!—006!)z=0: In this form it coincides with Ex. 1, 
Class 1, and its solution is z=Ae”*®*. 
Now A is an arbitrary function of y; call it f(y): then z=e”** f(y), where 
e” => represents an operation on / (y). 

But since ty rkasys Ze ks 
=(14+40+ he.) f(y) 
ey) 
it is evident that z=f(y+02). 
‘ d>z dz 
Ex. 2 wae a dy’ 
This equation may be written (d}—a 0) z=0. 
aera 
Now ve y dwe@-” -J/a (GREGORY'S Lxamples, p. 499.) 
vy co 
Vref ae ae ae ev *™ Fy) 
9 
=y Zdwe *ee2X" sy) Gt o=aVzd) 
