298 On Mr. Woolhouse’s Theory of Vanishing Fractions. 
dition; and if we take the expressions which are immediately 
deduced in the investigation, viz. 
a(a—b) , &(a—d) 
aoe? Y= ae? 
we see that they become vanishing fractions in the case of the 
circle, and do not limit the required point.” Now, I submit 
that the values 7 =a 4/1, y = 3, are the true, and the only 
values, fairly deducible from these vanishing fractions ; and that 
the fact of the problem admitting multiple solutions, under the 
proposed change of hypothesis, is altogether deduced from 
other, and distinct, considerations. It is, in fact, information 
which the analytical result is quite incompetent to supply ;- and 
is derivable solely from an examination, not of the conclusion, 
but of the original conditions of the problem. From this ex- 
amination it appears that, in the proposed hypothesis, the two 
conditions merge into one, and thus a restriction being re- 
moved, the problem becomes indeterminate; but the mere 
merging of the final result into the form 2, could never have 
made known this ; the information is obtained quite independ- 
ently of the slightest reference to this result, and from a di- 
rectly opposite source. It is no doubt true, that when condi- 
tions disappear, in certain hypotheses the results will assume 
the form 2, but it is not true conversely; that when the results 
assume the form 2, conditions must have disappeared, and 
thus the values of ° become innumerable, as Mr. Woolhouse 
contends, What would my friend say of the sum of a geo- 
a(r"—1), 
r—] 
his second proposition above, this sum is anything! It ap- 
pears to me that Mr. Woolhouse’s oversight, in his interpre- 
tation of ° in the foregoing problem, is analogous to that 
sometimes committed in physics; and which consists in taking 
for cause and effect, two phenomena, not invariably con- 
nected, yet both having a common antecedent. The very 
thing (viz. the hypothesis a = 5) which causes 2° to become 
9, causes also, in this case, one condition to disappear; and it 
is thence presumed that there is an invariable connexion be- 
tween these two events; whereas that connexion is purely ac- 
cidental. 
Belfast, March 16, 1836. 
a 
metrical series, viz.S = when7 = 1? According to 
