MR. W. SPOTTISW 0 ODE ON MULTIPLE CONTACT OP SURPACES. 
237 
Now the general term of □ ,m U is 
n(n— 1) . . (n— m+l)(— ) ; - 
i(m— 1) . . (m — i + l) 
1.2 
(02)” l “ i (01) i 0 B- ”T 
and it will be found on examination that the general term of □ " □ ,m U differs from that 
of □ ,m U only in respect of the factor (12)(m— 2i). Similarly the general terms of 
□ " 2 □ /m U, . . □ ,,p D /m U differ from that of □ ,m U only in respect of the factors 
(12) 2 (m— 2i) 2 , . . (12) p (m— 2i) p , respectively. 
This being so, let ; & 1 =d? 1 c)*+ . . , . . ; and let it be understood that 5 n o 2 affect 
the external subject of operation alone, then we may write 
□ ,m U = (02&j — 01S 2 ) m U ; 
and if we write the above expression in the form of a quantic, thus, 
□ te U=(l, 1, . .)(02i 1 , -01^ 2 ) m U, 
we may at once write down the expression for □ " p □ ,m U thus, 
(12 ?(mr, (m—2) p , (m- i) p , . .)(02^, -Ol^U (5) 
Having thus exhibited the form of the function □ l,p □ ,m U, we must next calculate the 
effect of the operation □ ' upon this function ; and if for this purpose we operate with 
□ ' upon the general term of □ " p □ ,m U, we shall find for the general term of 
□ ' □ " p □ fm U the following expression : — 
■ • (n-m)(-y(12) p f m{m 1 ) ^ (m-i+l) j 
!• ( 02 )“-‘ + 1 ( 01 )‘ 0 ”- 
=n(n— 1) . . 12) v 
m{m— 1) . . {m—i+ 1) 
' 1.2 . .i 
\(m~i+l)(m-2i) p +i(m-2i+2) p \(02) m - i+1 (01) i Q n - m - 1 l m - i+1 2 i . 
As regards the coefficient between the brackets { }, let m— 2^+1=^; then for 
])=1, 2, . .we have successively 
(m—i+l)(f*—l) +i(p+l) =mp>, 
(m—i+l)(ft—l) a +i(p+l)?=(m+l)fA a —2p, a -{-(m+l) 
=(m—l)p 2 +(m+l), 
(m — i + 1 )((&■ — l) 3 + ® ( p + l) 3 = (m +1)^ 3 — 3/a 3 + 3 (m + 1 > — 
=(m— 2> 3 +(3m+2)^, 
(m— i+l)(fA— l) 4 +i(p+l) 4 =(m-}-l)[/S— 4^ 4 +6(m+l)^ 2 — 4^ 2 -f(m+l) 
=(??i — 3)p/-b2(3w^-bl)jM< 2 -"]-(w2/-j-l) ’ 
2x2 
