236 MR. W. SPOTTISWOODE OM MULTIPLE CONTACT OF SURFACES. 
□ ,,2 D , U=0. Moreover 
□ '□"□'U=<12)n , (02.0 B - 1 l+01.0»- 1 2) 
l)(12){(02) 2 0 ra_2 l 2 — (01) 2 O ra-2 2 2 }, 
□ " □ f2 U =%(w-l)n ,, {(02) 2 0 n - 2 l 2 -2 .01.02. 0” _2 12 + (01) 2 0 Jl-2 2 2 j- 
=2n{n- 1)(12){(02) 2 0”- 2 1 2 — (01) 2 0”- 2 2 2 f; 
so that we have generally the relation 
□ '□''□'u=in"n' 2 ij, 
and consequently 
□ 3 u=n' 3 u+fn"n' 2 u ( 2 ) 
Proceeding to the fourth degree, we obtain, by means of the formulae already estab- 
lished, the following expression 
□ 4 U=: □ /4 U+ □ " □ ,3 U + f □ ' □ " □ ,2 U + f □ " 2 □ ,a U. 
But operating with □' on the expression given above for □ "□' 2 U, we obtain 
,2 U= 3n(n- (02) 3 0"- 3 l 3 - 01(02) 2 0 re " 3 l 2 2 
— (01) 2 02 . 0 re-3 12 2 -j- (01) 3 0 re_3 2 3 [ ; 
also 
□ ,3 U = n(n-l)(n-2) { (02) 3 0 re - 3 l 3 -3(02) 2 01 . 0”- 3 l 2 2 
+ 3(01) 2 02 . 0 ra - 3 12 2 -(01) 3 0 re - 3 2 3 [. 
Hence, operating upon this with □ ", we should find 
f □ ' □ w □ ,2 U = □ " □ ,3 U ; 
and consequently, collecting the former results, 
□ XJ=D'U, 
□ 2 u=n' 2 u+n' , n , u, 
□ 3 U = □ ,3 u +f □ " □ ,2 U, 
□ 4 u=n ,4 u+|n ,, n ,3 u+fn' /2 n ,2 u (3) 
Having now seen, in the special cases m= 2, 3, 4, the nature of the transformations 
contemplated, we may now proceed to the more general case ; viz. since the order of the 
operations □ □" is not indifferent, we have in general for □ m U an expression of the 
form 
□ m U = □ te U 
+ n , 'n' m - i u+n'n"n m “ 2 u+ . . 
+ W 
and our first object is to show how this may be reduced to the form 
A □ te U+B □ " □ ^u+c □ " 2 □ ,m - 2 U + . . A'n te - 2 U+ . . 
