380 
It may be interesting to inquire what relation this equation bears to 
the corresponding partial differential equation (IV); in other words, 
- whether there is any such relation between the two, as would justify us 
in denominating (V), 7x any case, the singular solution of (IV): what 
analytical conditions would be requisite in order that the former equa- 
tion should satisfy the latter: and what may be the geometrical signi- 
ficance of singular solutions of partial differential equations of the second 
order, if such singular solutions are admissible or conceivable. 
As regards the analytical conditions specified, they may be investi- 
gated thus. Differentiating the equation 
SF (2, Y, %, 4 B, y, 6, €) =0 
with respect to all the variables, we get identically 
GE 2 +(F) ay +2 aa a+ 1g 18 + Fay a d+ a =o, 
which, in consequence of the relations supposed, reduces to 
(2) a +(Z)ay <0 
or, as dx and dy are independent, we have the first two of the relations 
stated in the fifth article, namely, 
af af ie 
(a) " a YF 
So far all is plain; but when we proceed to differentiate these equations 
again with respect to all the variables, we get 
= z Z d d d\/d 
(Go) ae+( Fr) a+ (aa Zs dp mile teat a )e)- 
(22) ay «(2A) 20+ (ant + ap atte 43” a at)(F)=0, 
which are not equivalent to the remaining relations of the fifth article, 
namely, 
(G2)-». (2) (22)-0 
unless, simultaneously, 
d d ad d ad Of A 
(ae 7,4 8 apt ras dd —.+ ae +) ; ca 
d d d d df 
