5384 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
Examples. 1. Congruency of Bitangents to a Torse. (§§ 17-19.) 
§ 17. The simplest examples are equations of CLAIRAUT’S form, such as 
Y) = Pye + pe + Pips + Po: @ 
Yo = P2t + PP. + Po 
The complete primitive is 
yy = Oe + Oy" + yey + Co, (B) 
Yq = Cyt + C,CQ + C4" 
The singular solutions are given by the equations 
0 = (a + 2c, + €,) de, + (c¢, + 1) dea, | s 
0=c,de,+ (x +e+ 20) de, J mma yeti e y). 
Hence, by eliminating «, 
(G.de, —.c, de.) (de, -- de.) des eee eee) 
This is also of CLAIRAUT’S form, and its integral is 
(l—p)Q—pe =e. PONE A Ci Die tceriiels gol, (e), 
so that 
de, dG, =: yu: ln. 
Thus 
(2 + 2c; + &) (1 =p) +e (4 +1)=0, 
24 = —w—p-et+ pa, 
We eS yw oa 
26, = B — PX, 
4y, = (uw — &) (uw? + 3p + & — ap), ee ae (O 
dy) = — pw (p — a 
The equations (¢) furnish the first singular solution. For the second we must 
make (e), as a quadratic in p, have equal roots, that is, we must put 
