ME. W. SPOTTISWOODE ON THE CONTACT OE SURFACES. 
269 
And if we regard these equations as determining a particular direction for the axis, they 
express the condition that it must coincide with the tangent line to the curve of inter- 
section of U and V at the point P ; so that in this particular case the equations (35) 
do not involve ^ 1 V=0, . . as a consequence. 
Again, in the case of three-pointic contact, we may take the following form, viz. 
A 2 ^V+B 2 ^V-f . .+2BC&AV+. . = 0 (37) 
Then, since the operation i.ib l -|-A 2 J r ?oo 3 vanishes identically, we obtain, by operating with 
it upon u, v, iv respectively, and then eliminating u, v, w, the following resultant : — ■ 
i;V, 
$AV, 
*AV, 
-0 
.... (38) 
&AV, 
ajv. 
KKv, 
M.V, 
*AV, 
But this is the condition that (37) may be resolved into linear factors. Supposing it so 
resolved into the product (AP + . ^(APj-p. .), then one of these factors must vanish in 
virtue of (37). If, then, the contact subsists for three positions of the cutting plane, we 
may write 
AP+..=0, AjP-f . .=0, A 2 P+..=Q;;. ..... (39) 
to which we may add, in virtue of the identical equations, 
'?i 2 6iV+'U 2 ^Y+- . + 2mo^ 3 Y -f-. .=(wP + . .)(?;?! + . .) = (), 
w 2 t$fY+ b 2 c> 2 Y+. . + 2mYo 3 Y+ . . = (uV -j- . .)( ^P L -f- . .) = 0, 
the following, 
uF-\~. . = 0, wP-b..=0; 
(40) 
whence, eliminating P, . . , we obtain 
u , 
u, 
A, 
A;, 
a 2 , 
=0, 
V, 
V, 
B, 
B a , 
IV, 
IV, 
c, 
c 15 
c 2 
(41) 
showing that if the planes all intersect in a tangent to the curve of intersection, the 
conditions c$ 2 V=0, § 2 V = Q, . . are not of necessity fulfilled. 
It is perhaps unnecessary to pursue this part of the subject further. 
Returning from this digression to the equations (18), it may be observed that if there 
be two-pointic circumaxal contact about the point P, i.e. when nY=0, 0^=0, the 
equation (ra' □ — z3-n') 2 V=0 will be satisfied by tv T o values of m' : m ; in other words, the 
curve of intersection of U and V will have a double point at P, and along each of the 
branches the contact wall be three-pointic. Similarly, if there be three-pointic circum- 
axal contact about the point P, i. e. when in addition to the former (□V=0, r]'V:=0), 
we have □ 2 Y= 0, ( □ ' □ + □ □ ') Y= 0, □ ,2 V = 0, then the equation {*} □ — m □ ') 3 V= 0 
will be satisfied by three values of zu ' : ; that is, the curve of intersection will have a 
2 o 2 
