:me. w. h. l. eussell on the calctjltjs of symbols. 
257 
This product, when developed, will be of the form 
Then ®„(‘r) is given by the following rule : — 
Write down the following symbolical product : 
fli(7r+m)^2(T+m)53(5r+m) ; 
take every possible combination of the quantities 1, 2, 3,.. .w taken {n — m) at a time, and 
substitute them as the weight * of 5 in this continued product, diminishing [m) in each 
factor by the increment of the weight of the factor ; add all the results together, and we 
obtain the value of The truth of this rule will be manifest to every one who will 
consider the following result obtained by actual multiplication ; — 
4-3)+53(7r +2)+54(7r+l)+55(^)} 
+3X2(‘^+3)+5i(’r+3X3(^r-f 2)+S2(’*^+2)S3(’^“h2)-f-5j(‘r-f-3)94(‘r 4-1) 
-|-2)4-5i(’r -{-2)S2(‘^ +2)54(t + l)+5i(’r+2X3(7r+lX4(^+l) 
~l"^2('^+l)^3('^+l}^4(’!^+l)+^i('^+2X2(‘r+2X5T-}-^i(7r + 2 )^ 3 ( 77 + 1)^577 
+ ^ 2 ( 7 ^ + 1 + 1 )^5’r + ^ 1 ( 7 ^ + 2 )^ 477 ^ 577 + ^ 2 ( 71 ' + 1 )^47r^57J' + 037r6^7r6^7r } 
+ P { ^,(’*' + 1 )^2(7r + l)^3(7f + 1 )^4(7r + 1 ) + (77 + 1 )4(77 + 1 )^3(77 + 1 )^ 5 X 
+ ^4(77 + l)^2(7r + l)^47f ^57r + (tT + 1)^377^477^577 + S27r337794775577} 
+ ^{Trd^Trd^Trd^TrC^Tr. 
I now come to the investigation of the two new forms of the binomial theorem as 
explained in the former memoir. 
It is e\ident, in the first place, that in multiplying any binomial (f^+fX'^))” 
f^+fX"^)? the result in this case will be the same whether we employ internal or 
external multiplication. 
Let 
(e+e«MT=f"+f'-%.-,M+e"-"p2.-.^+ ■ ■ •. 
where ?>2„_i(77), (Pan-ii^), <P2n-3{'^) are unknown functions of ( 77 ) which we seek to deter- 
mine. 
Then multiplying externally and internally by 
+r<p2„-2(7r) 
-\-f^*^6{w-\-2n) -{-f%7r+2n—l)p2n-M+ • • • 
+ ^""+’f)(77) +^"”^7792„_,(7r + l)+ .. . 
* Tlie use I have here made of the term ‘weight’ will be familiar to every one who is conversant with the 
modem Higher Algebra. 
