ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SUEEACES. 
243 
Again, operating with □ = □'+ □" upon the expression for □ 4 U in terms of □ 
we obtain 
□ 5 u=n , 5 u+n ,/ n , 4 U+2n , n"n , 3 U4-2n ,, 2 n , 3 u+fn'n" 2 n' 2 u+-|n ,, 3 D' 2 u 
= □ ,5 U+f □ " □ M u +!□ 1,2 □ ,3 u +f □ 113 □ ,2 U +f (12/ □ ,3 U. 
But 
□ "□' 4 U=:(12X4, 2, 0, -2, — 4)(02^, -01d 2 ) 4 U 
=2(12)(2, 1, 0, -1, -2)(02^,-01^) 4 U, 
□ ' /2 D' 3 U= (12/(9, 1, 1, 9)(025 1? -015 2 ) 3 U, 
□ " 3 □ ,2 U=(12) 3 2 3 (02 3 Sf— 01 2 ^)U, 
which, in virtue of the condition □ 3 U= 0, 
= -(12) 2 ^(02^-0B 2 ) 3 U; 
hence 
□ 5 U= (02^ — 01S 2 ) 5 U 
+ 10(12X1, 2, 0, -2, -V) (02§j, -01S 2 ) 4 U 
+^(12/(3, 1, 1, 3^ (02&„ -01S 2 /U 
— 4(12/(02^— 01& 2 ) 3 U 
+1(12/(02^ -01& 2 ) 3 U. 
But collecting the terms of the third degree, we have for the coefficient of (12/ 
3-4+fX02) 3 ^+(-^+12-f • 3)(02) 2 (01.)i?i 2 +. . j-U 
=20((02) 3 ^ 3 — (01) 3 ^)U ; 
so that, finally, 
□ 5 U = (02^ — 01S 2 /U 1 
+ 10(12)(l,2,0,-2,-35(02S„-0iyu . (2) 
+20(12/((02) 3 & 3 — (01) 3 &|]u, J 
which, for greater symmetry, may be written thus : — 
(1,5,10,10, 5, 1^ (02^— 01S 2 ) 5 
+ 10(12X1, 2, 0, -2, -1^ (02^ — 01S 2 ) 4 
+20(12/(1, 0, 0, 1^(02^- 01S 2 ) 3 . 
These expressions may be rendered somewhat more compact by writing as follows : — 
B=0 n-1 2S 1 +0 n-1 & 2 ;} ^ 
2 L 
MDCCCLXXVI. 
