ME. W. SPOTTISWOODE ON THE CONTACT OF SUEFACES. 
273 
so that the whole expression sought 
= . .)(«, v, w) 2 
~(v 1 w l -{-w 1 v 1 —2u'u', . .){u, v, w)(u, v, iv) 
-f ■ (^i Wi — w' 2 , .. )(w, wf} 
= — w 2< I> suppose. 
This being the case, the solutions of the equations 
{hi 3 - ci. 2 y v = o, (c\- a i 3 yv = o, (<a - lyyv = o 
may be written in the following forms : — 
hl-y — clW=±uc Y — T>, —^-uh Y — O, 
<A 2 V— a&JV=±w « — O, =^vc\/ — <P,-. . 
alW—hliV=±wh ■/ — <X>, ==F 
• (51) 
which involve <X> — 0. <f> is therefore a surface which cuts U in a curve, at each point of 
which there is an axis, 
a:h: c=l;V : ITV : l 2 V, 
about which there is tliree-pointic contact. 
It may be shown also, by the following geometrical construction, that if tliree-pointic 
contact be triaxal it will be superficial. If we take three axes, PQ, PQ„ PQ 2 , and draw 
through each three planes ; then if three-pointic contact subsist for each triplet of planes, 
the contact will be circumaxal for each axis, and therefore triaxal. If we take a fourth 
axis, PQ 3 , the following planes will pass three and three through each of the axes, and 
will serve for the planes required, viz. the planes 
PQjQa, PQ 2 Q, PQQ 1? PQQ 3 , PQjQg, PQ, 2 Q 3 , 
say the planes (A, B, C), . . (A s , B 5 , C 5 ). Taking the forms Ac^-f B£ 2 +CA for □, the 
conditions for three-pointic contact along each of these planes will be 
(Ac) -j-B § 2 + C <5 3 )A =0, (AgO; -j- BA -j- C£)°\ =0, | 
(AA+BA + CA)-Y=0, (AA + BA+CA)A=0,[ (52) 
(a 2 s,+ba+ca) ! v=o, (a s s,+ba+Wv=o. j 
Eliminating d,V, & 2 V, . . ^AV, . . , we obtain the resultant, 
A 2 , B 2 , . . B C , . . 
A 2 , B- ..BA,.. 
A 2 , B 2 , ..BA, •• 
(53) 
which is the condition that the six planes should all touch a cone of the second degree. 
But the planes in question pass three and three through four lines ; and as it is impos- 
sible through any one line to draw more than two planes touching a cone of the second 
