554 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
The line 
i On meine ates so 2 ((B), 
Yo = Cyt + db, i 
will be an inflexional tangent if the equations 
WiL== 0, Chile = 0,  eMLiGn? = 
are satisfied together, where 
M= $ (, eye + d,, Cope + 0°). 
By eliminating w and putting p, for c,, y, — px for b,, &., we have two 
equations of CLAIRAUT’S form satisfied by the lines of the congruency. 
§ 42. For the singular solutions we have 
wile, dh, = 0. Mee tar (4). 


ade dbs = .0' oy, (oh i ss ee re) 
= ite des + 55 db, + 5), ds =a ©), (because Fi = 0) nen, (5), 
eM ooM o7M OM OM. \ 
ened de, + ene. dey + ay db db, + anal dos — 0) (because eva a 0) sana) 

aye ee, 1 aurea, + ap vas, 2 + 5.35 ate t ae =O rere 
But 
Oe, a i Ob, ‘ Oc, 7 B Ob, 
and (7) gives 
cM eM OM oM 
eS ak aa 5, des +} ae de, + 5, de, = 0. 

Therefore 

