FROM THE DIFFERENTIAL EQUATION. 
339 
in the first members of which z is supposed to be expressed 
in terms of x, y, p, q by means of the differential equation, in 
the second members, in terms of x, y, a, b by means of the 
complete primitive. 
Now the singular solution is deduced from the complete 
primitive by means of the equations 
dz 
da 
=-0, 
(27); 
and it is evident from the form of (26), that this will generally 
involve the conditions 
dz 
dp 
= 0, 
dz 
dq 
= 0 
(28). 
Such then will generally be the conditions for determining 
the singular solution from the differential equation. 
The conditions (28) will not present themselves, should the 
denominator of the right-hand members of (26) vanish identi 
cally. But it may be shewn that in this case the conditions 
(27) do not lead to a singular solution. And analogy renders 
it probable that whenever the conditions (28) are satisfied the 
result, if it be a solution at all, will be a singular solution. 
The complete investigation of this point, however, would in 
volve inquiries similar to those of Chapter vm. 
The Buie indicated is then to eliminate p and q from the 
differential equation by means of the equations (28) thence de 
rived. 
11. The following geometrical applications are intended to 
illustrate the preceding sections. 
Ex. 1. Bequired to determine the general equation of the 
family of surfaces in which the length of that portion of the 
normal which is intercepted between the surface and the plane 
x, y, is constant and equal to unity. 
As the length of the intercept above described in any sur 
face is z (1 +p 2 + <f) h , we have to solve the equation 
s 2 (1 +p 2 + q 2 ) = 1 (u). 
22—2