244 
ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SUEEACES. 
and referring to the developments given above it will be found that 
□ 2 U = 0A 2 U + (12)BU, "j 
□ 3 U=0A 3 U-f(12)BAU, I 
□ 4 U=S 3 A 4 U + 6^(12)BA 2 U + 3(12) 2 (A 2 + B 2 )U, w 
□ 5 U=Q 2 A 5 U+10S(12)BA 3 U + 5(12) 2 (A 2 +3B 2 )U. _ 
For three-branch contact at P the number of equations, in addition to those for ordi- 
nary contact, is three ; and replacing the suffixes, we may take for these equations 
the following, 
□ ? a U=0, □? 3 u=o, n; 4 =0; 
or, transforming as above, 
0A? 2 U + (12)B 12 U=O,d 
0A? 3 U+(13)B 13 U=O, l (5) 
0A? 4 U + (14)B 14 U=O. J 
Now since the surfaces are supposed to touch at the point P, we shall have 
0=01 : 0” _1 1 = 02 : 0 B “ 1 2 = .. 
If they touch also at the point P„ we shall have also 
10 : l n_1 0=12 : l n ~'2= . 
and so on for other points; so that when the surfaces touch at the point P„ we shall 
have 
J 
( 6 ) 
when they touch also at the point P 2 we shall have 
0 : 2 ,l_1 0=21 : O*-^ . 2- 1 lO 
0 : 2”- 1 0=23 : O"-^ . 2»" I 3, > (7) 
... • • • j J 
and so on for other points of contact. 
These equations show that if the surfaces touch only at the point P, there is no 
means of eliminating any of the ratios 0 : 12, 0 : 13, . . If, however, they touch also at 
a second point P 1; we can eliminate all the ratios of the form 0:12; i.e. those contain- 
ing the symbol 1 in the denominator. If the surfaces touch at a third point P 2 , we can 
eliminate all the ratios of the form 0:1 2, 0:13, 0:23; viz. all those containing either 
of the symbols 1 or 2 in the denominator. It may be remarked that when the surfaces 
