1911-12.] Singular Solutions of Partial Differential Equations. 163 
Hence, moving from S along the tangent plane to the osculating 
surfaces, we reach a onefold infinity of other corresponding points 
SiS 2 S 3 .... and from each of these we may proceed by drawing new 
integral elements. Thus, commencing at S, we trace out a surface which 
is touched at each point by two integral surfaces, and corresponding to 
every one of the onefold infinity of points S'S" .... on the unodal line 
a similar surface can be constructed. 
Finally, therefore, we trace out a onefold of surfaces, touched at every 
point by two integral surfaces, and these constitute the onefold of singular 
solutions. 
The unodal lines 
h = 0 , if/y + bif/ z = 0 , 
will then be the loci of the points of contact of parallel tangent planes, and 
will therefore be isoclinal lines for the onefold in question. 
The unodal lines being given by 
a= -'f'x/if'z, b= -xj/y/ i]s z , 
the direction cosines of the normal to the onefold are 
• -'I'y/'l'z : “ b 
and the singular onefold is therefore obtained by integrating 
- i^dx - i^dy - dz = 0, 
giving 
if/(xyz) = c. 
It follows that if we regard the normals to the integral surfaces at any 
point in space as forming a cone, each such cone has a double generator. 
{Issued separately May 17 , 1912 .) 
