342 SYMMETRICAL AND MORE GENERAL SOLUTION 
and this represents various systems of cones and other develop 
able surfaces. 
Similar but more interesting applications may be drawn 
from the problem of the determination of equally attracting 
surfaces. 
12. Attention has already been directed to the different 
forms in which the solution of a non-linear equation may 
sometimes be presented. It may be added that linear equa 
tions admit generally of a duplex form of solution. The ordi 
nary method gives directly the equation of the system of 
surfaces which they represent; Charpit’s method leads to a 
form of solution which exhibits rather the mode of their 
genesis. 
Ex. Lagrange’s method presents the solution of the equa 
tion 
(mz — ny) p + fix — lz)q — ly — mx (a), 
in the form 
lx + my + nz = (f) {x 2 + y 2 + z 2 ) {h), 
the known equation of surfaces of revolution whose axes pass 
through the origin of coordinates. 
Charpit’s method presents as the complete primitive of (a) 
(x — cl) 2 + {y — cm) 2 + [z — cn) 2 — r 2 (c), 
c and r being arbitrary constants. This is the equation of 
the generating sphere. The general primitive represents its 
system of possible envelopes. 
These solutions are manifestly equivalent. 
Symmetrical and more general solution of partial differential 
equations of the first order. 
13. The method of Charpit labours under two defects, 
1st, It supposes that from the given equation q can be ex 
pressed as a function of x, y, z,p; 2ndly, It throws little light 
of analogy on the solution of equations involving more than 
two independent variables—a subject of fundamental import 
ance in connexion with the highest class of researches on 
Theoretical Dynamics. We propose to supply these defects.