Remarks’ on Professor Wallace’s Reply to B. = 303 
in which case there is in fact nothing produced but the iden- 
tical equation 1=1 or O=0. 
This introduction of m=0, in its vanishing state, is 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 @ logical point 
of view, in this application of the method in the paper Vol. 
VII. page 284. Other similar objections might be made, but 
it is unnecessary to extend these remarks any farther. Fe 
Demonstration of the Binomial Theorem for integer positive 
values of the exponent, referred to in the former part of this 
papers 
If we multiply 1-+-x by itself and that product by 1+ 
and so on, we shall ively obtain (l+a2)*=1+4+2r+ 2, 
(1-+2)*=1+ 3c2+327+0%, (14+x)*=1+ 4a + 607+ 42° 
+-x*, all of which are contained under this general expres- 
sion, 
—] nn—ln—-2 
(4oy-ltpetyg ty gg ett Be (A) 
which by the above multiplications is true when n=1, n=2, 
n==3;n=4. To prove it to, be true for all integer positive 
values of n it is only necessary to show that the multiplica- 
tion of the formula (A) by (1+) will produce a similar ex- 
pression Composed in n+ 1 as that formula is in n, or that 
n+i n-+1 n-+1 n+inn—I 
ae au 
for from thence it would follow that being true forn=4, it must 
be true for n+1=5; being thus true for n=5, it must be 
true for n-+1=6, and so on ad infinitum. Now by perform- 
ing the multiplication of the series (A) by !+<, and placing 
the products by 1 and by x beneath each other according to 
the powers of x, it becomes, by introducing into the lower 
3 4 
line the factors + 23 4° Se. instead of 1, for the sake of 
symmetry, and including in parentheses the equal factors of 
the two products : 
pe 
