520 Prof. Young on the Theory of Vanishing Fractions. 
analytical results” accompanies his animadversions at page 
398-9 in the last number of this Journal; he appears to think 
it sufficient that the antecedent equation should be satisfied, for 
he remarks, ‘* The corresponding antecedent equation to the 
result z = 2, when cleared of fractions, is 07 = 0, or 0 = 0, 
an equation that is obviously satisfied without any limitation 
to the value of z, and that cannot fail therefore to be compa- 
tible with the other equations or conditions.” The statement, 
in connexion with this remark, viz. that “‘® can never be the 
symbol of absurdity,” has a little surprised me, because the 
contrary is a fact so generally known to analysts. To occupy 
these pages by examples of this would be quite superfluous, as 
they abound in most of the Continental books on algebra. In 
the comprehensive work of Bourdon there is an ample supply of 
such examples, and from which he deduces the ordinary conclu- 
sion, viz. that “le symbol © est tantot un caractére d’indéter- 
mination, tantot un caractére d’absurdité.” 
From what has now been said of the symbol 2, it appears 
that, when it is not the indication of absurdity, or of incompati- 
ble conditions, it may arise from either of these two causes: viz. 
i ‘ Sie Je 1° 
from taking the ultimate, or limiting, value of =, the general 
result of an analytical process; or, without regard to this ex- 
treme limit, it may arise from the destruction of one or more 
of the conditional equations. One or other of these circum- 
stances must take place in connexion with the occurrence of 
5 whenever this symbol is at all interpretable. I say when- 
ever the symbol is znterpretable, for cases may arise in which 
this symbol is indicative of neither multiple solutions, nor of li- 
miting values, nor of incompatible conditions. In such cases 
therefore other modes of solution must be sought. ‘The in- 
stances to which I now allude are among those in which the 
vanishing of the numerator is not necessarily accompanied by 
the vanishing of the denominator ; but where each vanishes in- 
dependently, in virtue of distinct hypotheses introduced among 
the arbitrary quantities in each. With the exception of these 
unintelligible results, the occurrence of © is always traceable to 
one or other of the circumstances before mentioned; which 
circumstances, although having no necessary connexion, may 
nevertheless, as in the case of the ellipse question, both exist 
simultaneously. 
When therefore 5 takes the place of =, in any hypothesis, 
Be 
ii P 
we may be assured that the limiting values of Q will always 
. 
