264 
MR. W. SPOTTISWOODE ON THE CONTACT OE SURFACES. 
those of — Y in ci ; , we may write = e>;U. Secondly, the result of any combined operation 
such as oo t U consists of two parts, say V (or ^V), where the accented operators affect 
Y only, and (or B"^'U), where the doubly accented operators affect U only. 
But the conditions c).y = 0, o 7 -V = 0 give, as is well known, ~d x Y 'b ;/ V = 0v , . ., where 6 
is indeterminate ; so that if, after the operations ojoj have been performed on U, we replace 
d x V by 6u, ~d,(Y by 0v, we shall have an ' expression identical with that obtained by 
evaluating h'fi'jXJ. In other words, 
when ^y=0, 2i/V=0, then o/pV^o^y (19) 
Similarly, if, in addition to o t \ r =0, ^V=0, we have 
then 
S 2 V=0, ^V=0, l]Y=Q, 
v s? v=o, ^v=o, 
^ v=o, ^v=o, 
••• ^v=wv=o. 
.-. %y=^;y, >. 
ofo/)jV = o/yjY. _ 
( 20 ) 
( 21 ) 
The equations (IS) may be regarded as equations involving an unknown quantity us' : us, 
which determines the section along which there is a contact of a given degree. Thus, 
in order that the surfaces may have ay-pointic contact at the point P along some section 
through the axis, we must have 
(ot'd — ^□ , ) p - 1 V=0 (22) 
But, inasmuch as whenever there is a yepointic contact along any section there must 
also be a p — l,y> — 2, . . pointic contact, it follows that, in addition to the condition 
above written, we must also have 
(us' □ — ^□ , ) p - 2 V=0, (us' □ — us □ , ) 7 ' _3 V=0 . . ; . . . (23) 
and the conditions for the existence of a y-pointic contact at the point P along some 
section through the axis will be expressed by eliminating us' \us from the equations (22), 
(23), taken two and two together. 
Thus the condition for a three-pointic would be 
( □ 'V) 2 □ 2 V— □ y □ y( )v+ (□ y) 2 □ ,2 y == o, . . . (24) 
or 
(□ 2 v, (□ □ , + n , n)y, □ ,2 y)(n'A T , — □v^ 2 =o (25) 
Similarly, for a four-pointic contact we should have, in addition to (25), the following : 
(□ 3 V, (□'□"+□ □ '□ + n 2 n')V, (□' 2 n + n , n □'+□ n' 2 )y, □ ' 3 V)(m'v, — nV 5 3 =: 0 ,( 26 ) 
and so on, for higher degrees, the azimuth of the plane section being always deter- 
mined by the value of □ 'Y : □ V at the point P. And it may be observed that, under 
the conditions supposed, there will in general be only one plane section along which the 
contact will subsist. 
