SIR B. C. BRODIE OX THE CALCULUS OF CHEMICAL OPEKATIONS. 797 



A consequence of this commutative property is that 



{xy)''=ary''; 

 for 



V {ccyY=xijay 

 =xxyy 



Further, let tliere be a compound weight V, of which is the symbol, of such a nature 

 that it is identical with the component weight A without the component weight 13, and 



let - be selected as the symbol of the weight Y. Then 



y 



whence, on similar principles, if ^be the symbol of a compound weight V identical with n 



component weights A, «j component weights Aj, n.^ component weights A2, without 



m component weights B, Wj component weights Bj, m^ component weights B^ , 



6=- 



1 2 



where x, a'l, Xo, . . • • y, y^, y.^, ■ ■ • • are the symbols of the weights A, xij, A2, . . . ., 

 B, Bj, Bg . . . . respectively. 



We may also reason thus: xy is the symbol of a weight which results from the 

 successive performance upon the unit of space of the operations y and-Tj^^ is the symbol 

 of a weight which results from the performance of these operations in an inverted order, 

 and (xy) is the symbol of a weight which results from their joint performance. Now, 

 since the result is the same in whatever order the operations be performed, and since it 

 is immaterial whether the operations be performed jointly or successively, we infer that 



xy=yx=(xy). 



(6) If in the composite symbol (p one of the factors be the symbol of a group, so that 



(f)=x(y+yi), 



<p is to be interpreted as the symbol of the weight which results from the combination 

 of the weight A with the group of weights B and B,, the group being collectively con- 

 sidered and as constituting a single weight (Sec. I. Def. 4, 5, and Sec. II. (4)), — not, 

 however, be it observed, a single weight compounded of the weights B and B,, which 

 would be symbolized by yy^, but a single weight constituted of B and Bj which is sym- 

 bolized by {y+yi). (Sec. I. Def. 4, 5, and 7). Now the weight A will be combined with 

 the group of weights collectively considered, if it be combined with the constituents of 

 the group severally, but the symbol of the weight A combined with the two constituents 

 of the group severally is xy-\-xy-^. We hence arrive at two symbols for the same weight, 

 which express indeed two different aspects of the same object, but which are identical as 

 regards the object signified ; whence 



MDCCCLXVI. 5 Q 



