162 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
and hence the necessary and sufficient conditions for a Cauchian Singular 
Solution are 
+ P<t>z =6, <t> y + q<j> z = 0. 
By a similar process exactly we can deduce the geometrical interpreta- 
tion of the Cauchian Eliminant, corresponding to various singular loci on 
the isoclinals. In particular, consider the case where each isoclinal 
possesses a unodal line ; the system will then be given by equation (15). 
If 
A = -tyx/l {/ z , B = -if/y/lf/z, 
the differential equation obtained by substituting p for a , and q for b in 
Fig. 2. 
(15), will be the general form of an equation with a onefold of singular 
solutions 
i l/(xyz}.-c. 
Each isoclinal surface, therefore, has a unodal line given by 
if/ x + a\j/ z = 0, if/y + bif/ z = 0. 
We may suppose every such surface in the neighbourhood of the 
unodal line SS'S" .... to be formed by a system of curves PQRST, 
P'Q'R'S'T' .... etc. (fig. 2). 
Across PSQ and RST a system of integral elements pass, and through 
S there are two, which therefore touch, and have in general different 
curvatures; the same is true for S'S" .... and intersecting as they do 
on the same isoclinal, the tangent planes at SS'S" .... are all parallel. 
Round S there is a onefold of other points SjS 2 S 3 .... say, lying on 
neighbouring unodal lines, at which the integral elements differ infinitely 
little in direction from that at S. 
