398 Mr. Woolhouse on the Theory of Vanishing Fractions, 
anotion. On the other hand, it is rather remarkable that 
Professor Young did not consider that 3 was the usual symbol 
of absurdity or of incompatible conditions, and that § could 
never be so, in the result of an investigation logically con- 
ducted. Thus, the corresponding antecedent equation to the 
Oo adie hs 
result x =o, when cleared of fractions, is oz =0 or 0 =0, 
an equation that is very obviously satisfied without any limi- 
tation to the value of z, and that cannot fail therefore to be 
compatible with other equations or conditions; but the cor- 
responding antecedent equation to the result «= 2 is: =m. 
an equation evidently indicating the presence of absurdity or 
of incompatible conditions, unless the nature of the inyestiga- 
tion will admit of infinite results. 
The query respecting the geometrical series is dismissed at 
once by a reference to the fourth extract from my essay. By 
putting for S the series it represents, the equation is 
9 n—1 a (r” — 1) 
atar+ar-+.....+t ar o> Sree 
and as the left-hand member is not discontinuous when 7 = 1, 
the vanishing fraction, which forms the right-hand member, 
must be limited to its continuous value, viz. 2a. The very 
circumstance of the equation involving both a determinate 
and an indeterminate quantity, when r =1, indicates the ex- 
istence of a fallacy in the process by which it has been de- 
duced. We first have 
SS a ar a7? foes 27 pec0sn(G) 
and multiplying by r — 1, we get 
(ry —1) S=ar*—a =a (r®— 1) seveee (8) 
which divided by 7 — 1, gives 
S = oe aa eoeeeesessce (c) 
In the case 7 = 1, and ry — 1= 09, we have therefore com- 
mitted the fault of multiplying by absolute nought in passing 
from (a) to (6); but the equation (c) is a true deduction from 
(b), for the mere placing of r— 1 in the denominator of a. 
fraction is not an actual performance of division. The equa- 
tion (a) becomes S =a +a +a + «3 the equation (4) 
entirely vanishes, and (c) becomes S = = 
After the foregoing discussion it will be needless to offer any 
special observations on the obvious inaccuracy of Professor 
