338 
DERIVATION OF THE SINGULAR SOLUTION 
whence b = — a. Tims, the values of a ancl b being variable, 
but such that b is a function of a, the proposed solution is 
a particular case of the general primitive. 
Some general questions, but of minor importance, relating 
to the functional connexion of different forms of solution, will 
be noticed in the Exercises at the end of this chapter. 
In quitting this part of the subject, we may observe that 
there are two inodes in which the questions it involves may 
be considered. The first consists in shewing that the gain 
of generality, which in Charpit’s process accrues in the tran 
sition from the complete to the general primitive, is equal to 
that which Lagrange’s original but far more difficult process 
secures by the employment of the general value of p drawn 
from (4), instead of a particular value drawn from its auxiliary 
system. The proof of this equivalence, as developed with 
more or less of completeness, by Lagrange and Poisson, 
[Lacroix, Tom. u. p. 564, III. p. 705), and recently by Prof. 
De Morgan, [Cambridge Journal, Yol. vn. p. 28), is, from its 
complexity, unsuitable to an elementary work. The other 
mode is that developed in the foregoing sections. 
Derivation of the singular solution from the differential 
equation. 
10. The complete primitive expresses z in terms of x, y, 
a, b. The differential equation expresses z in terms of x, y, 
p, q. Either is convertible into the other by means of the 
two equations derived from the complete primitive by differ 
entiating with respect to x and y respectively. Hence it is not 
difficult to establish the two following equations, 
dz 
ddz 
dz ddz 
dz 
da 
dbdy 
db dady 
dp 
ddz 
cfz 
d' l z d~z 
dadx 
dbdy 
dady dbdx 
dz 
ddz 
dz d^z 
dz 
da 
dbdx 
db dadx 
dq 
Jz 
<fz 
(Cz (Pz 
dcidx dbdy dady dbdx „ 
(26),