ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OF SUEFACES. 
239 
Again, 
1 □ " 2 n ,m U=n □ //» □ Im+lJJ _|_ (12)2 □ /m+iuj 
_2l_ n//2nta+2U I m ) 2 ( — -+lll=l' m+2 U 
— (to + 2)(to + 1) U U + 
whence the system 
□ „ □ » □ ">P= n □ " □ M u+ ^ (12f D'»«U, ) 
/to — 1 
□ f3 □ 1,2 □ ,m U = 
(to + 3) (to + 2) 
□ " 2 □ lm+3 TJ (12) 2 □ lm+s V, 
> • 
( 8 ) 
Assuming this to be true for m x — 1, we should have 
□ /»!-!□ Hs n f»U— 
’(to + TOj — 1)(to + TOj— 2) 
then operating again with □ we obtain 
to(to— 1) ,,2 , m + m r ITT ■ ( m l ) m n 9\2(-| 
_9'l U U U “T TO + TO) — 2 ' ' ^ 
□ Inn n 112 n /)»U : 
TO (TO — 1) 
D „ 2D , m+mi u 
' (to + TOj— 1) (to + TOj — 2 
, / i n 2 )2cr--»-»,u 
1 (to + TOj — 1 ) (to -L TOj — 2) 'to + TOj — 2j ' ' 
and the numerator of the expression within the brackets 
=m 1 m(m+TO 1 — 2) ; 
so that the expression sought finally becomes 
n ,,2 n ,m+m iTJ— — — (12) 2 n' m+mi U (9) 
(to+to 1 -i)(to+to 1 -2) u u to+toj- i" ' u u, . . . 
which proves the law. 
Once more, operating with □ ' upon the expression given above for □ ' □ " 3 □ /m U, we 
obtain 
□ » □ »> n'~V=~ {£=i □ "■ □ '- +! U+^2 5 (12)- □ " □ '-«u| 
3ro + 2 to + 1 ( 12 )2 n » n to+2 U 
' to + 1 m + 2 v ' 
whence the system 
□ -. DMn ^ U= ^^“^L . a "» D ^ U+ ^ | . g+^) i - (12)- D » D '-U ; | (10 
□ ■< D " D ^p = (w ”^p 1 »”p 2 ) 2) . D >"D^. u + ^^+yp% ) -(i2)»n"n^u. j 
