Remarks on Professor Wallace's Reply to B. 303 



m which case there is in fact nothing produced but the iden- 

 tical equation 1 = 1 or 0=0. 



This introduction ofm=0, in \{^vanishing &\.2^ie^\s what Berke- 

 ley particularly objects to ; calling such an expression the 

 '^ Ghost of a departed quantity^ From this example it is 

 evident that there is no peculiar excellence, in a logical point 

 of view, in this application of the method in the paper Vol. 

 VII. pa^e 284. Other similar objections might be made, but 

 it is unnecessary to extend these remarks any farther. 



Demonstration of the Binomial Theorem for integer positi'Ve 

 values of the exponent^ referred to in the former fart of this 

 paper. 



If we multiply 1+ a; by itself and that product by l-{-a) 

 and so on, we shall successively obtain (l-f-x)- = J -f-^a;-f-a;^, 

 (l-fa;)3 = l + 3a; + 3a?^4-a;'% (l-fx)* = l+4a; -{■ Qx^ + Ax'-" 

 -{-a;*, all of which are contained under this general expres- 

 sion, 



n n n—l n n — I n—2 



(i+xy=i+Y^-^i'~2~"''-^i'~2 — ^^'+ ^c- (A) 



which by the above multiplications is true when n=l, n=2, 

 n=3, n=4. To prove it to be true for all integer positive 

 values of w it is only necessary to show that the multiplica- 

 tion of the formula (A) by (l-f-a;) will produce a similar ex- 

 pression composed in n+1 as that formula is in n, or that 



?i+l n-\-l n-{-l n n-\-lnn—l 



(l+x) =l-\-~ — a;+— j— .qa^^-f-^-- — g--a;3-f-Sic. 



(A') 

 for from thence it would follow that being true for ?i=4, it must 

 be true for n-\-l=5 ; being thus true for n=5, it must be 

 true for n4-l=6, and so on ad infinitum. Now by perform- 

 ing the multiplication of the series (A) by \-\-x, and placing 

 the products by 1 and by a; beneath each other according to 

 the powers of x, it becomes, by introducing into the lower 



12 3 4 

 line the factors -• gi - 7' &ic. instead of 1, for the sake of 



symmetry, and including in parentheses the equal factors of 

 die two products ; 



