me. w. spottiswoode on multiple contact of suefaces. 
245 
touch at the three points P, P„ P 2 , the ratio 0 : 12 may be eliminated by either of the 
two formulae 
tf: 1-0=12: 0-1. 1-2, | 
0 : 2”- 1 0=21 : 0*-*2 . 2* _I 1. J 
That the conditions so obtained are equivalent to one another, and not independent, 
is both obvious a priori and is capable of being shown by multiplying the denomi- 
nators of the first of the equations by 2 n_1 0 and those of the second by 1" -1 0. The 
equations then take the form 
0 : l"->0 . 2”' 1 0=12 : (p-T . I*- 1 2 . 2 S ~ 1 0 
=21 : 0" -1 2 . 2 W_1 1 . 1— J 0. 
But since the surfaces touch at the three points P, P 4 , P 2 , the last two denominators 
are in virtue of (18) of § 1 equal. Hence each one of the two equations (8) involves 
the other as a consequence. 
Returning to the equations (5) and eliminating the ratios 0 : 12, . ., we obtain the 
three conditions : 
1 b- 1 0A? 2 U+0* -1 1 . l n-1 2B 12 U=0,"l 
1” _1 0A? 3 U + 0 ,l-1 l . 1" -1 3B 13 U=0, S> (9) 
l-'OA^U+O- 1 ! . R- 1 4B 14 U=0 .j 
If the surfaces touch at a third point P 2 , we may replace the equation □ ? 4 U = 0 by 
□ 2 3 U=0. If they touch also at the point P 3 , the results of elimination may be put into 
two forms, viz. 
2»- 1 0AI 3 U+0 w - 1 2 . 2»- 1 3B 23 U=0, S-'OJ&U + O""^ . 3"- 1 2B 23 U=0, 1 
3 b_ 1 0A 31 U + 0 n-1 3 . 3”' 1 1B 31 U=0, l-^AJJJ+O- 1 ! . 1*- 1 3B S1 U=0, > . (10) 
l»- 1 0A? 2 U + 0 ,, - 1 l . 1»- 1 2B I2 U=0, 2 ll - 1 0A? 2 U+0 n - 1 2 . 2“- , lB ia U=0. J 
If the surfaces touch only at the three points P, P x , P 2 , we shall have only the 
first form of the first equation, the second of the second, and either form of the third. 
If the surfaces have three-branch contact at a second point P l5 we may derive the 
conditions to he fulfilled, by a similar process, from the system 
□ *U 1= 0, 0^1=0, □ * i U 1 =0, 
and so on for any number of points. 
For four-branch contact the number of equations is four, which may be written thus : 
4A? 2 U + 3(12)B 12 A 12 U = 0, ^ 
^Af 3 U + 3(1 3)B 13 A 13 U= 0, 
4A? 4 U + 3(14)B 14 A 14 U = 0, j" 
A^U + 3(15)B 15 A 15 U=0;j 
2 l 2 
(H) 
