268 
ME, W. SPOTTISWOODE ON THE CONTACT OF SUEFACES. 
ditions for universal two-pointic contact. Hence if a two-pointic contact subsists for two 
positions of the cutting plane about an axis, it will subsist for all positions about that 
axis. It will be shown in the sequel, as is well known from other considerations, that 
under these circumstances the contact will hold good for all axes through the point P. 
A similar result follows in the case of three-pointic contact. If the third equation of (18) 
holds good for three values of m ' : sq, say ^ : cq ; m ' 2 : w 2 ; sr 3 : ra- 3 , then writing down the 
equation for the three values successively, we shall be able to eliminate the three 
coefficients of the powers of nr' : cr and obtain the resultant, 
f 2 t 2 
'oJ l 7v n G7 1? 
'2 t 2 
W ^ CO 2 (O 2 j 2 } 
/ 2 f 2 
, ST 3 7tT 3 , To., 
= — (sqcq SqTO\j)(tqCTo ) == 0 , (34) 
which cannot be satisfied, since by hypothesis the three values of : gj are all different. 
Hence the coefficients of the equation in question must separately vanish. In other 
words, if a three-pointic contact subsist for three positions of the cutting plane about 
an axis, it will subsist for all positions about that axis. 
The same law may obviously be extended to contacts of higher degrees. 
The axis may be drawn, as before stated, in any direction through the point P ; it 
may therefore be made to coincide with a tangent to the curve of intersection of U and 
Y at the point. But in that case it is obvious that two-pointic contact would subsist 
Tor two positions (in fact for all positions) of the cutting plane without involving the 
conditions for the ordinary contact of the two surfaces (viz. ^Y=0, e5 2 V=0, o 3 V=0) as 
a consequence. It is perhaps desirable to show that the formula; here employed take 
cognizance of this circumstance, as w r ell as of the corresponding circumstances in the 
cases of contact of higher degrees. 
Suppose, then, that two-pointic contact subsists for two positions of the cutting plane 
about the axis, say for the two planes (A, B, C, D), (A„ B n C n D,) ; then, adopting the 
last form of the group (8), we have the two equations 
A&! V +Bc 5 2 V T-C<h V=0,) . 
aav+bav+cav=o.J - (oj; 
Adding to these the two identical equations, 
wdjU +«;d 2 V -f-wi 3 y=0, 
«o t U -fto 2 V + wo 3 Y = 0, 
and eliminating ^V, & 2 V, c$ 3 V, we obtain the resultants 
= 0. (36) 
