Prof. Young on the Theory of Vanishing Fractions. 519 
that “ Professor Young involves himself in a palpable incon- 
sistency when he arrives at the fact of the ellipse question 
admitting multiple solutions, by an examination of the origi- 
nal analytical conditions, and at the same time alleges that the 
analytical result is quite incompetent to supply that informa- 
tion.” The mathematical readers of this Journal will however 
readily perceive, that what is here,charged as “ palpable in- 
consistency” is in perfect accordance with the strictest analy- 
tical accuracy ; and that the “inconsistency” would have been, 
in inferring the multiple solutions from the analytical result, 
without reference to the original conditions, as Mr. Woolhouse 
has done, thus assuming (what is not true) that the converse 
of a certain proposition holds merely because the proposition 
itself is known to be true. Mr. Horner in the present volume 
of this Journal (p. 43.) has brought forward whole cluster 
of instances, in each of which, as he clearly shows, “ the ana- 
lytical result is quite incompetent to supply the information” 
even as to whether the question admits of a single solution, 
much less as to whether it admits of multiple solutions: the 
information sought must be obtained in all these cases, as I 
have obtained it in the ellipse question, viz. by a direct appeal 
to “ the original analytical conditions.” Without such an ap- 
peal how are we to know whether the analytical result to which 
the condition 
Qa4+Vx2?7—T=5 
leads, viz. 
32°— 202+ 32=0, 
will supply values competent to satisfy that condition? The 
presumption is that it w2// supply such values; upon trial how- 
ever we find them to fail: and yet these values will satisfy the 
immediately antecedent equation, but this is not sufficient ; 
every anterior step must be satisfied, up to the original equa- 
tion inclusively ; and the error committed in overlooking this 
would be precisely similar to that which Mr. Woolhouse ap- 
pears to me tohave committed, in inferring the multiple solutions 
to the ellipse question, merely because these solutions satisfy 
the final result*. ‘The same mistaken view of the “theory of 
* It is but justice to Mr. Woolhouse to state, however, that he admits 
(p. 399) that “ the nature of the problem, as originally presented, is the pro- 
per source of rejective information,” although he maintains that the original 
analytical conditions do not furnish the proper source of information, as to 
whether, in certain hypotheses, one of those conditions becomes destroyed, 
or two or more of them become dependent ; but, on the contrary, that the 
Oo. , side a . 
result 5 8 asufficient indication that one or other of these circumstances 
must take place. (See III. p. 394.) Ihave endeavoured to show, however, 
that this result is not competent to furnish any information on the subject. 
