240 
ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SUEFACES. 
Assuming this to be true for m x — 1 , we should have 
□ f»,-i n // 3 n f»u= 
m[m— 1 ) ( m — 2 ) 
+ — 2) (m + wq— 3) 
(I 2 ) 2 n ,, n ,m+mi - 1 ir ; 
then operating with □ ' we obtain 
□ lm x □ 113 □ fi»U — 
(m + mj— m 
m + m \ 3 //3 / m+TOl -y 
m + mj 
(l 2 ) 2 n"n' OT+mi U 
(l 2 ) 2 n' / n' m+m ‘U 
m(m — l)(m — 2 ) 
□ " 3 □ te+m >U 
m 
(l2) 2 n"n' m+m 'XJ 
|(m— l)(m— 2 )( 3 m+ 3 m 1 — l)( 3 m 2 -f 3 mm x — 7 m— 2 m I + 2 )(m+m 1 — l)j- ; 
and the quantity within the brackets { j- 
= 3 m,(m— l)(m— 2 )+( 3 to— l)(m— l)(m— 2 ) 
{m x (%m— 2 )+( 3 m— l){m— 2 )){m x (m x +m— 2 ) — (m— 1 )}, 
of which the part independent of m x obviously vanishes, and the remainder 
—m x {?>(m— l)(m— 2 )+m 1 ( 3 m— 2 )(m x +m— 2) — ( 3 m— 2 )(m— l)(m— 2 )(m x +m— 2 )| 
or if we put m-\-m x — 3 =^, 771+^— 2 =^+ 1 , m x =p— {m— 3 ), the above expression 
becomes 
=?w 1 {3(to— l)(m— 2)+(/a — 777 + 3)(3777— 2)(|K> + 1)— {Zm— 2)(m- 1) -j-(3m — l)(m— 2)(f*+l)} 
=m 1 {3(7»-l)(7w-2)-(7»-3)(3»i-2)-(3m-2)(m-l)+(37»-lXw-2) 
+ jW ,[-(3TO-2)(m-4)+(377z-lj(77i-2)]+^ 2 (377i-2)} 
= ^ 1 (772-}-77i 1 — 3)1(7^-!-^! — 3)(377i — 2)~l-7772, — 6} 
= m x (m -f 777 -! — 3 ) { 3 t 77 2 + ( 37 ^ — 4 )t n + 2 m x } ; 
so that the expression for the value of □ ,B1 i □ " 3 □ ,ro U finally becomes 
( 11 ) 
(i 2 ) 2 n"n' OT+OTi U, 
which proves the law. 
It seems unnecessary to pursue these developments further. 
