ME. W. SPOTTISWOODE OX THE CONTACT OF SUEFACES. 
271 
two axes passing through P (say PQ, PQ,), and a pair of planes passing through each 
(say PQQ n PQQ 2 , and PQ^, PQiQ 2 ) ; then, if two-pointic contact subsist for each pair 
of planes, the contact will he biaxal, as was shown at the commencement of the present 
section. We shall now have three planes in all, PQ,Q 2 , PQ 2 Q, PQQ, (say the planes 
A, B, C ; A,, B„ C, ; A 2 , B 2 , C 2 ), forming a solid angle ; and in virtue of the equation 
with which we started, we shall have 
A^Y+B^V+C^Y=0, j 
AAy+BAV+C^ 3 y=0, 1 T (4G) 
AAV + B A V + c a V = 0 . ) 
But as these planes by hypothesis do not pass through one and the same straight line, the 
determinant of these equations cannot vanish. Hence the system (46) can hold good 
only on the conditions ^,V=0, <$ 2 V = 0, cijV— 0. But we may take, as before, the 
tangent to the curve of intersection at P as one of the axes PQ, PQ t . Hence we come 
to the same conclusion as before. 
Passing to the case of three-pointic contact (and supposing that two-pointic superficial 
contact subsists at the point P), and equating to zero the coefficients of the powers of 
vs' : vs in the equation (vs' □ — ar[I]') 3 V = 0, and adopting the same form as before, we 
shall obtain 
a 2 r;V+3 3 S 2 V+ . . +2j9y&AV+ . . =0, 
aa'($?V+03'S 2 V + . . +0V-f-3V)^AV+ • . =0, - . 
«' a ijv+0p s*v+ . . +20VUV+ . . =0, 
(47) 
which, by means of the transformation (16), may be reduced to the following forms: — 
(ti 3 -bl 3 )*V = 0, (cA-^,) 2 V = 0, (b\-al. 2 yV= 0, ' 
(dAj — A)(^l— «^)V = 0, 
(b\-ab 2 )(ci 2 -b\)v=o, 
(cA-^K-cSJV-O, 
whereof three only are independent. 
And if the contact be triaxal, we should have (taking the first of these forms) 
c 2 c> 2 V — 2b c &AV+ W=0,' 
c$V- 2J ,<^AV+ WV" = 0, 
ay - 2 AcMv + = o . J 
(49) 
Eliminating the coefficients, we obtain, by the usual method, 
(b 1 c 2 ~b 2 c l )(b,c—bc. 2 )(bc l — b 1 c)-0 (50) 
But as by hypothesis the three axes are all distinct, this equation cannot be satisfied ; 
and therefore (49) can coexist only on the conditions 5 2 V=0, £AV=0, § 2 V=0. Plence 
