238 
ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OF SUEFACES. 
(m— l) p +«(^+l) p =(w— 
+ JT 2 T 3 (3m-^+5y~ 2 
+ ^-iK y -^- g (6m ^+ 9y _ 4 
+ • • 
1 - 2.. 7 
in which it is to be understood thaty is always an odd number. 
From these transformations we obtain the following results, viz. 
□ □ ,m u=^ T n"n /m+1 u, 
m + 1 
□'□' /2 n ,m u = m ~ ] □ " 2 □ te+i u -K12) 2 □ ,m+i u, 
171 + 1 1 \ / 
mi 9 I 9 
□ ' □ " 3 □ ,m U □ " 3 □ ' m+1 U + C12) 2 □ " □ ' m+1 U, 
m + 1 1 v ' m + 1 ’ 
□ ■' □ "p □ '"'U = m 1 □ " p □ ,m+ i jj + y p - c? ■ ^ — Hur 5 (i2) 2 n'' ? - 2 n ,,n+i u+ . 
m+l 1.2.o wi + 1 v ' 1 
> ( 6 ) 
+■ 
+ 1 ‘1.2.3 m + 
p{p— 1) • • {p ~i + 2) jm—p + 2j—\ 
(i2y- i n^-' +i n te+i u+ . . 
1.2 . .j m + 1 
But since the general term of the expansion of IH m U will be of the form 
□ " p □ lb □ " a □ /<7 U, it follows from these formulae that every such term, and consequently 
the whole expansion, may by successive steps be reduced to a series of terms of the form 
□ l'p-9 Q ' 
Before proceeding further with the main question it will be worth while to notice a 
few consequences of these reductions. Thus, in the first place, 
□ » □ " □ ”V=~ □ ' □ " □ '" +, U=^1 • ~D " □ ■ M U ; 
whence we obtain the following system : — 
m + 1 
□ ,2 n"n ,m u : 
n' s n"n' m v-- 
.□"□' m+2 u, 
n t mi D „ n n // n 
m + m x ) 
