1911-12.] Singular Solutions of Partial Differential Equations. 161 
gian Singular Solution of (17), being derived by exactly the same process; 
moreover, the latter is touched at any point by some member of the two- 
fold (17). 
Suppose PKR 1 R 2 SS 1 S 2 (fig. 1) to be any surface (ab) touching E the 
envelope at the point P, then we may suppose the part of this (ab) isoclinal 
near P to be formed by an infinite number of curves RPS, R^PS^ R 2 PS 2 , . . . 
all touching E at P. Each of these lines is cut at all points right up to 
P by parallel integral elements, which coalesce at P without in general 
crossing- E. 
Now all round P there is a onefold infinity of points P'P" .... at 
E 
Fig. 1. 
which the envelope E is touched by neighbouring isoclinals, whose integral 
elements are slightly inclined to those of RPS. 
Hence the integral surface enveloped by the elements of corresponding 
points on the PP'P" .... isoclinals converges to P and stops, forming a 
conical point at which the tangent cone has collapsed into a line. 
Hence the Cauchian Eliminant is in general a locus of such points. 
If the direction of the envelope E is also the direction of the integral 
0 ^, 0 ^ 
surfaces, then E is the envelope singular solution. Now ^ and ^ for the 
envelope are given by 
0.*'^ 0 S = O 
OX 
and 
0„ + ^0i==O, 
ay 
VOL. XXXII. 
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