534 
ON SINGULAR SOLUTIONS. 
[545 
Writing moreover 
disctp cf) (x, y, p) = 0 
to denote the equation between (x, y), such that for any values of {x, y) which satisfy 
the condition, or say for any position of P on the p-discriminant locus, there is a 
two-fold value of p. By what precedes, it appears that the ^-discriminant locus is 
made up of the envelope locus, cuspidal locus, and the tac-locus ; as I infer, each of 
them once. 
The foregoing are the abstract principles : I consider the singular solution to be 
that given by the equation which belongs to the envelope-locus (viz. I do not recognise 
any singular solution which is not of the envelope species) ; and the result of the 
investigation is, when we seek in the ordinary way to obtain the singular solution, 
whether from the integral equation or from the differential equation, that we account 
for the extraneous factors which present themselves in the two processes respectively. 
I reserve for another communication the discussion of particular examples.