ME. W. H. L. EUSSELL ON THE CALCULUS OE SYMBOLS. 
261 
Let us assume 
_l_(Aayn-i_^A(’^^^™-^+Al'y''-^+&c.)7r 
_l_ ( Af &c. y + &c. 
_j_ ( + AiY"-^ + A^Y”"" + &c. y+ &c. 
Then we have, writing ^„(t) for 1,2.3 ....m ’ 
A®=2^(2w) 
Aw=2^(2w-1)2^(2%) 
Al"’ = lD{2n-2)^(}{2n-l)2d2n 
&c. = &c. 
Ai“^=2^(2w-s+l)2^(2?i-s+2) .. . . 2^2?i 
A[l]=25,(2n) 
A1‘) = 2^(271- l)2^i(2n) + 2^.(277- l)2^(2w) 
Ai'>=2^, (271-2)23 (271-1)2^ (27i) 
+ 2^ (27i-2)S^i(27^-l)23 (27i) 
+2^ (27i-2)2^ (27^-l)2^.(27^) 
&c.=&c. 
A<'^=2:^.(27z-s+ 1)S^ (27i-s+2) . . . ^^(27 i) 
+^^(271-5+1)2:^.(271-5+2) . . . ^0(2n) 
+ ... 
+X6{2n—s+l)t6 (271-5+2) . . . ^^,(271) 
AW=S^2(27i) 
Ai^'>=X^,(2n-l)X^2n+tO,(2n—l}t^,(2n) 
+X0 i2n-l)XU‘^n) 
Af>=S^,(27i-5+l)2:^ (271-5+2) . . . X6(2n) 
+X6 {2n-s+l)XU^n-s+2 ) . . . X^(2n) 
+ ... 
+Xd {2n-s-\-l)X6 (271-5+2) . . . X6l2n) 
+S^.(27^-5+l)2:^.(27^-5+2) ...t 6 (271) 
+ 2^.(271-5 + 1)S^ (271-5 + 2) . . . S^.(27 i) 
+ ... 
5 1 
Where there are (5) terms in the first part of this expression, and s • in the second : 
AW=2^..(2^-«+1)^^^.(2»*-s+2)S^,.(27i-5+3) . ..t6^i2n) 
+ 2:^J271-5+1)S^,^(271-5+2)S^,/271-5+3) . . . S^,/27 i) 
+X^^(27i-5+l)^^^X2w-s+2)S^i27i-5+3) . . . t&,i2n) 
+ ... 
2 N 
MDCCCLXII. 
