ME. W. SPOTTISW OODE ON MULTIPLE CONTACT OE SUEFACES. 
229 
This result may be generalized by introducing operative symbols, as follows : — 
Let P 10 =57 1 
be a differential operator, capable of operating on functions of x, y, z, t, and containing 
x\, z i; ty as mere constants ; then 
□ 12 (or more completely expressed f~°~| 12 ) 
— p y p _p y p 
is a differential operator capable of operating on any function of # 1? y^ z„ ; x 2 , y 2 > tl ; 
in which, when Y is a quadric in x, y, z, t, P 10 V=P 01 V 1 . Operating with □ 12 on 
U = (x,y, z, t) n , we have □ 12 V=a function of the form 
Cl, y» «i, ti) fa, y 2 , z* 4) (#, y, z, t) n , 
linear in the coefficients of V, and also in those of U. 
With this explanation we may write 
01 . — 02 . 0' 1_I 1= □ , 2 U, 1 
01.0 B - 1 3-03.0 w - 1 l=n 13 U, j 
>- 
02.0 ra " I 3— 03.0 re - 1 2=D 23 U, 
or, more briefly. 
0" -1 l, 0 n ~% . . 
01 , 02 , . . 
= n 12 u, n 13 u, . . n 23 u, 
( 9 ) 
( 10 ) 
and the system (5) may then be replaced by the following, viz. 
□ 12 u=o, n 13 u=o, . . n 23 u=o, (ii) 
any two of which may be taken as the conditions required. Similarly the system (6) 
may be replaced by the following, viz. 
□ ? 8 U=0, □? 3 U=0, . . □ 12 D 13 U=0, (12) 
any three independent members of which may be taken as the conditions required. 
If the two surfaces touch at a second point P„ we may form expressions similar to 
(10) but involving the coordinates of P 3 in the place of those of P, thus : — 
l-ffi, l n ~% . . 
10 , 12 , . . 
— □ 02 U 1 , CIlosUu . . D 23 U„ 
(13) 
and the conditions for contact at the point P 2 will be comprised in the system 
□ 02 U 1 = 0, □o.U^O, . . □ 23 U 1 =0 (14) 
Similarly, the conditions for contact at a third point P 2 will be comprised in the 
system 
□ „U 2 =0, □ OS U S =0, . . □„U,=0, (15) 
and so on for any number of points. 
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