516 Prof. Young on the Theory of Vanishing Fractions. 
In my former communication, I contented myself with sim- 
ply pointing out the fallacy involved in the extremely general 
statements which I extracted from the Essay referred to; and 
with tracing the source of this fallacy to the circumstance of 
the author having unguardedly assumed the converse of a cer- 
tain proposition, to be equally general with the proposition 
itself, which converse holds however only in particular cases. 
The direct proposition to which I here allude is this, viz. 
that when in certain hypotheses any of the analytical conditions 
of a problem disappear, the final result, to which the general 
process leads, takes the form 2. The converse proposition is, 
that when the final result takes the form © original conditions 
must have disappeared. ‘This latter is the affirmation distinctly 
conveyed, without the slightest qualification, in the propositions 
marked II. and III. in Mr. Woolhouse’s reply; and it will be 
remembered, that against those propositions only my objec- 
tions were directed; for.I cheerfully admitted that much of 
Mr. Woolhouse’s Essay was ‘in strict accordance with the 
usual notions of this doctrine.” 
To show that these objections were valid, I adduced an in- 
stance (that of a geometrical series) in which the propositions 
objected to would lead to error; and in adverting to this in- 
stance, in his reply, it will be seen that my respected friend 
has not defended the positions in question from the charge of 
making the sum of the said geometrical series anything, but 
has shown that another position (Prop. IV.), a position which 
was never impugned, is competent to supply the correct result. 
Surely my ingenious friend does not consider it to be a suffi- 
cient defence of Proposition III. to prove that its affirmations 
are neutralized by Proposition IV.; and yet there is no other at- 
tempt made to establish its truth. The proposition which Mr. 
Woolhouse discusses at page 395, does not at all contribute to 
this object; for that is the converse of the one which it behoves 
him to prove, in order to establish his third principle: this 
principle requires the proposition stated above, in Italics, and 
not the one which Mr. Woolhouse has demonstrated in the 
preceding Number. There is no dispute as to the form of the 
result when conditions vanish; the question is, does this form 
necessarily imply vanishing conditions in the original analytical 
statement of the problem? Mr. Woolhouse’s third principle 
unequivocally states z# does. But innumerable examples to 
the contrary may be adduced. The well-known problem of 
Clairaut, which has for its object the determination of the spot 
between two lights, which is equally illuminated by both, is a 
case in point, and furnishes a satisfactory refutation of the 
