1911-12.] Singular Solutions of Partial Differential Equations. 159 
The importance of this will be seen not to be overestimated if it is 
remembered that it is really from among equations of higher orders that 
mathematics receives its most frequent application in physics. 
The singular solutions of partial differential equations of the first order 
have been fruitfully discussed by Lagrange and others from yet another 
point of view, viz. as regards their derivation from the complete primitive. 
The Lagrangian Singular Solution of the Partial Differential Equation 
whose twofold integral is 
f(xyzab) — 0 (13) 
is obtained by eliminating a and b between 
f~ 0 ) fa = 0 , f b = 0 . 
If \[s(xyz) = 0 be this singular solution, then it is easily shown that (13) 
is of the form 
/= (a - A) 2 i/q + 2(a - A)(b - B)if/ 2 + (b - B) 2 i ^ 3 + onf/(xyz) = 0 . . (14) 
where \Jr v \/s 2 , and \fs s are functions of xyzab, and A and B are functions of 
xyz. This enables us at once to verify that every member of (14) touches 
the singular solution at a point, if we remember that any point on it is 
given by the three equations 
if/ = 0, A = a, B = b, 
and \Js = 0, A = a meet (14) bis where B — b meets it, while \Js = 0, B = b meet 
it bis where A = a meets it. 
This has, however, been thoroughly discussed by Professor Hill in his 
paper to the Royal Society “ On the Locus of Ultimate Intersections of 
Lines and Surfaces,” 1892. 
In the special case where a = 0 in (14), 
/= 0, f a = 0, f b = 0, 
are satisfied by 
A = a, B = b, 
and there is no Lagrangian Singular Solution corresponding to these values, 
although there may exist one for other values of a and b. 
If yjr v \fs 3 are functions purely of x, y, and 0 , then there does not 
exist any Lagrangian Singular Solution at all. In all cases it can easily 
be shown that every member of 
(a- A)V x + 2(a - A(5-B)^ 2 + (6-B)V 3 = 0 . . . (15) 
has a unodal line given by 
A = a, B = 5. 
