387 
The solution of this equation by the symbolic method, in the form of a 
general primitive, is readily found to be 
LY U+4 y 
= 24 — b(a-y) +¥(@- y). 
The solution of the same equation, in the form of a complete primitive, is 
s= 4 {ac® + By? +(1-a+B) ay} +ye+dyte. 
These solutions may be readily identified by assuming 
P(e -y) =A(fe+y) +A’, 
V(a-y) = Bia - y)* + B’ (@—y) + B’, 
whence, by substitution in the former, we get 
Gi 2 
and, by comparison of co-efficients, 
4A +B =4a, B~ZA=438, 2B= 3(e+ f), 
ZA’ + B’=y,3A'- B=6, B= e. 
= L(A(@t_y)+A@+y)) + (B@-y)'+B(e-y) +B’ 
A similar identification may be performed upon the solutions, in the two 
forms, of the equation 
r+é=l1. 
Thus it appears that the general primitive may be reduced to the com- 
plete primitive, but not vice versd. 
(6.) The following will serve as examples of the latter portion of the 
third article. Let the partial differential equation proposed for solution be 
dw e dw a (aol “(Eye i ey i a "|, 
"ae Gy de \° \de dy ds } 
where a, 5, ¢ are given constants: then the solution of this equation, in 
the form of a complete primitive, is 
& 
w= ax + By + ye- {a™a™ + b™ B™ + c™ y™\" = 0, 
and the singular solution is 
Be 
Thus, if m = 8, the singular solution of the partial differential equation 
Be dao dips ae el my 3 al v al 
dic Y ly ede zt % da 
