MR. W. SPOTTISWOODE ON MULTIPLE CONTACT OF SURFACES. 
241 
It has now been shown that the expression for □ m may be reduced to the form 
n hn +pn"n" n -'+(yn"' 2 +'y'n")n m - 2 + . .( 2 1 *n" m " 1 )n ,a - 
Hence, on replacing each term of this series by its value given by (5), we find that 
the expression for □ m U may ultimately be reduced to the form. 
(A(02&„ -(m 2 ) m +B(12)(02^ -0B 2 ) m "’+ . . K(12) m " 2 (02^, -01^ 2 ) 2 }U, (12) 
or, still more symbolically, 
(A, B, . . K)((02&„ -0U 2 ), (12)) m - 2 (02§„ -01^ 2 ) 2 U, 
excepting in the case of m— 2, when there is one extra term, as was seen at the outset 
and as will be noticed again below. This being so, we may eliminate the quantities 
01, 02 by the formulae O1 = 0O” _1 1, O2=0O” _1 2, and then divide out (P throughout. The 
expression is then reduced to the following form, 
(A, B, . . K)(S(0"~ 1 2S 1 , -O-Hi,), (12)) m_2 (0" _1 2^, -O^T^U. 
But if the surfaces touch at either of the points P 1? P 2 (say PJ, we shall have 
12 — ^ i»- i 2 
^ Ul± 5 O’-TO- 1 ! 1 
6 : 1" -1 0=12 : O'- 1 ! . l-^ 
(13) 
so that the expression in question may be finally cleared of all quantities relating to the 
quadric V (01, 02, 12), and reduced to the form 
(A, B, . . K)(P , - 1 0(0 , - 1 2& 1 , 0 re -T . i-^)—^-^, -O^T^U, (14) 
in which it is to be remembered that S 2 operate only on the external subject U, and 
not upon any of its derivatives occurring in the operative factors themselves. 
It is in the eliminations effected by means of the formula (13) that the main difference 
between the methods of this and of the former papers consists. The conditions for 
multiple contact here established are more numerous, and at the same time of lower 
degrees, and therefore more stringent, than those found before; but they appear to 
carry the subject to it3 limit. 
If the surfaces touch also at the point P 2 , we may in like manner use as the formulae 
for elimination the following, viz. 
02 2™~ 1 0 
21=4,2 ,l -T 0 u a 
-L, O— 0 „-i 2 Qn-12 U Z- 
The results obtained by these two methods cannot of course be independent. In 
fact the equivalence of the two forms may be readily shown as follows. The first result 
is a rational function, homogeneous in the two quantities l n-1 0, 0 B-1 1 . l m-1 2, say 
(1” -1 0, (P-T. l»- 1 2) m - 2 =0; 
and in like manner the second may be represented by the equation 
( 2 »->o, o*-’2 . 2 W -T) TO " 2 =0. 
