556 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
In order to test whether this is a solution, we differentiate totally, multiply by 
1, ¢,, ¢,, and add, and we have 
d F) d\'/0 a ot le Oh 
ie + Pig + P25) i. ee toy, — ban) Oe B * Onn ap i Oy, = 
where the dashes outside the brackets indicate that in differentiation ¢, and c, are to 
be treated as constants, 
In this equation the substitution p, =¢,, p, = cy will make the coeflicient of x 
disappear, but the other terms will generally not vanish, and therefore the locus of 
parabolic points is not generally a solution of the differential equations of the 
congruency. 
It is, in fact, generally speaking, the cusp-locus of the first singular solution. 
§ 45. Suppose, now, that the surface has a cuspidal curve. It may be shown that 




this is a solution. 
For at any point (#, y,, y2) on the surface, the equations 
Op = Gee oD Y, = €,0 4-105: 
Mi; oM/op = 0, o°M /op? = 0 
are satisfied by putting 
[Pe bh =y — oe, by = Yn — Cy 
if ¢,, C, ave determined by the equations ° 
5 ASR 2 Ob Ob As 
ay, 1 Oy? ae 2 Oy, Oy, +e e Sy 


From the latter may be deduced, by differentiation, on the supposition that 
de = dy,/¢, = dy>/¢,, which is consistent with ¢ = 0, that 

eee 0 Og Ob Oo 
eee % e255) e+ 2(3 alee Bey qe e057) (a, ON + gat oa a) 
9 (22 Oey dc, \ / Op Od Op 
+2(= 1e Ghar. eG age liGee Fe Gray ° oe )= O.a eee (9). 

The equation (9) holds all over the surface. 
Now, at a singular point where there are two coincident tangent planes, 
