ME. W. SPOTTISW OODE OX TIIE COXTACT OF SUEFACES. 
if the contact be triaxal it will be superficial. But we may take for two of the axes of 
triaxal contact the tangents to the two branches of the curve of intersection through P ; 
and for every position of the cutting plane about each of these axes the contact will be 
three-pointic, viz. two consecutive points of the branch to which the axis is a tangent 
and one point of the other branch will lie in the plane ; whence it follows that, reckoning 
only arbitrary axes as before, if three-pointic contact be uniaxal it will be superficial. 
And this method has application to all degrees of contact. 
The equations (48) would apparently determine two axes about which three-pointic 
contact would be circumaxal ; but that this is not the case will appear from the actual 
solution of one of them. In fact the solution of the third equation depends upon the 
quantity (^ 2 V) 2 — in order to develop which we have the following values : — 
a?v=- 
V , , 
u' , 
v, 
+5 
Vi, 
t! , 
V, 
It', 
w„ 
tv, 
it!, 
tv i, 
w, 
v. 
tv , 
• 
v. 
tv , 
• 
i 
ii 
> 
tv , 
v' , 
u, 
+0 
tv', 
v' , 
u. 
It 1 , 
w„ 
w, 
it! , 
W„ 
tv. 
V , 
tv, 
• 
v , 
tv , 
• 
*sv=- 
It,, 
V', 
u. 
+ 0 
u„ 
v' , 
u , 
v' , 
tv„ 
W, 
v' , 
tv„ 
tv, 
u. 
tv , 
. 
u, 
tv , 
. 
Hence, by the method of compound determinants, in the expression (M 2 V) 2 — ^V^V, 
the term independent 
of (3 
=tv 2 
tt, , 
w' , 
v' , 
tt. 
— —w 2 (v l w l —i ! 2 , . 
•) {u, 
V, tv) 2 , 
tv' , 
A , 
tt! , 
V , 
v' , 
tt! , 
w„ 
tv, 
It , 
v , 
tv , 
• 
the coefficient of 6 2 
= vf 
tv'. 
v' , 
tt, 
= —w 2 (v l w ,~tt! 2 , 
.)(u, 
V, tv ) 2 ; 
tv'. 
v, , 
t! , 
V, 
v' , 
It' , 
tv „ 
IV, 
tt , 
■V , 
tv , 
• 
while the coefficient of Q will be found to be 
=iv 2 (v 1 w 1 -{-tv l v 1 —2u'u' i . .)(u, v, w) 2 ; 
