242 
MR. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SURFACES. 
On multiplying the first by (2" '0) OT 2 , and the second by (1” ’O)™ 2 , we obtain the two 
expressions 
(T'-’O . 2 a_1 0, 0" _1 1 . l n ~'2 . 2 n - 1 0) m - 2 =0, 
(T^O . 2»" 1 0, 0” -1 2 . 2 n ~ l l . l M " 1 0) m - 2 =0. 
But since the surfaces touch in the three points P, P 1? P 2 , it follows that 
(P -1 1 . l n ~ l 2 . 2 n - 1 0=0“- 1 2 . 2“ _1 1 . l s -'0. 
Hence the two expressions are equivalent. 
It is further to be noticed that the last term of the expression is of the form 
{ (0 2) 2 £ 2 + (0 2) 2 c5 2 } U, 
according as m is even or odd. Consequently from m — 5 and upwards the last term of 
□ m U may always be eliminated by means of the expression for □ ” l_2 U ; and the equa- 
tion finally depressed by one degree in 0 and (12). 
The expression (14) when equated to zero will form one condition, which must be 
satisfied, either by the coordinates of the points, or by the coefficients of U, in order 
that it may be possible to draw a quadric having (m+l)-pointic contact with U at the 
point P, and contact of the same or of lower degrees at other points P 15 P 2 , ... 
Such is the general theory; but the subject will perhaps become more easily intel- 
ligible by the aid of the next section, in which the cases of m— 2, 3, 4, 5 are examined 
in some detail. 
It will probably have been remarked that we have here developed only expressions 
of the form □ ™U, and have taken no account of those of the form □ p l3 □ i 2 U. But the 
latter, which would have been more complicated, are happily unnecessary ; since the 
eliminations above indicated will be always possible for expressions of the form □ ” l U = 0, 
provided only that one of the subscript numbers shall always correspond with that of 
one of the points at which contact takes place. And this may always be ensured, 
because in all the investigations of this paper, except those contained in § 4, contact, 
two?pointic at least, is supposed to subsist at more than one point. Thus, if there be 
contact at a second point P„ we may use the operators □ l2 , □ 13 , . . ; if there be contact 
also at P 2 , we may use also the operators 0 23 , 0 2i , • • ; and so on for any number of 
points. 
§ 3. Conditions for the cases n= 3, 4, 5, 6. 
With a view to examining more in detail the cases of m= 1, 2, 3, 4, 5, we may write 
down the developments indicated in the preceding section thus : — 
□ U= (02^-01^)11, 
□ 2 U = (02&j — 01§ 2 ) 2 U + (12X02^+01 & 8 )U, | 
□ 3 U=(02^-OB 2 ) 3 U + 3(12)(O2^-Ol^)U, i >(1) 
□ 4 U=(02^ — 01& 2 ) 4 U + 2(12)(3, 1, -1, -3)(02§ 1 ,-01^) 3 U+6(12) 2 (02 2 S 2 4-01^ 2 )UJ 
