54 PROFESSOR KELLAND ON A PROCESS 
Ex. 2. To solve the equation 
1. By the method employed at p. 269 of my previous Memoir, this equation 
becomes 
(—D+)y+a(—1) Dea eye 



/-D+3 
-+ 
br 92 Pee! ek 
x dx-* x x D—34 
_1 f{Xdz 
nae ab 
=P, suppose ; 
y il (J i i d> s) 

then a Pr os 
dx 
2 —a* bs 
y=Ane*4 re “fe de (Gets) 
dx  dxt 
2. By Professor Booue’s method, given in the Philosophical Magazine for 
February 1847. The given equation when written 
x(d—a d)y—ty=X2; 
may be thrown into the form 
f(@(xF @))y=X-«; provided 
f(a) F (d)=d—ad', and 
f' (@) F @) =-} 
We have, therefore, f()=(@—a)~ 
F (d)=(d—a)? ad? 
and y=d-* (d}—a)—? {a-} (dt—a) Xz} 
3. By Mr Harcreave’s method, given in the Philosophical Transactions for 
1848, p. 31. 
By changing d into 2, and 2 into —d in the equation 
z(d—ad*)y—ty=Xz.... (1) 
it becomes d (x—azt)y+iy=dX&.... (2) 
which, being an ordinary linear equation, gives, as its solution, if we write 
d-1 for ['dz, 
yaa} (zt =a)—2d-"{@t—a) dK} 2. . (3) 
Now, since equation (2.) has been derived from equation (1.) by the change 
of d into x, and x into —d, equation (1.) may be derived from equation (2.) by 
changing 2 into d and d into —#. Consequently (in some cases, at least, of 
which this form is an instance) the solution of equation (1.) may be derived from 
the solution of equation (2.) by the same change. Hence the solution of the given 
equation is 
