in reply to the Observations of Professor Young. 397 
which the expression (2) is deduced will consequently, when 
x = a, be also satisfied by any value of y._ If this primitive 
equation expresses an original condition of the problem, that 
condition, therefore, when x = a, cannot limit the variable y, 
or the expression (2), to any particular value. If, however, 
this equation is produced by a combination of two or more 
leading equations of the problem, the circumstance of its 
wholly disappearing when x = a, will necessarily lead us to 
the conclusion that for this particular value of 2 some depen- 
dency exists among those leading equations, and therefore that 
one of the original conditions of the problem becomes, in that 
case, virtually destroyed. In addition to this proof we may 
remark that the expression (3) is legitimately deduced from 
the equation (1) in every case in which x — a does not ab- 
solutely vanish, or in which the value of x differs from the 
quantity a, however small that difference may be; that since 
it holds good when z is taken as nearly equal toa as we please, 
and is in itself continuous as 2 approaches and arrives at that 
value, it is evident that, when x becomes exactly equal to a, it 
will express, as in (4), that particular value of y, or of the 
vanishing fraction (2), which unites in the law of continuity 
observed by all its other successive values. 
Having attempted, and I expect successfully, the demonstra- 
tion of the principles laid down in the extracts from my essay, 
without discovering them to be “ fallacious,” it now remains for 
Professor Young, since the truth is our common object, either to 
subscribe to my views or to point out wherein consists the inac- 
curacy of the reasoning here employed; and, without any wish, 
to prolong our discussion, I unhesitatingly pledge myselt to de- 
vote my most respectful and candid consideration to whatever 
arguments or explanations he may be pleased to offer. But it 
will be useless to pursue the subject any further unless Pro- 
fessor Young will enter more into the theoretical merits of the 
question and make up his mind to support every general 
statement with some kind of evidence. 
In Professor Young’s present letter he thinks it remarkable 
that I did not reflect that was as likely to be “ the symbol of 
absurdity” as the symbol of multiple values, and he follows up 
the same idea by observing that “* when we are operating with 
equations of the first degree, containing several unknown quan- 
tities, the symbol ® is, in fact, the very form which the result 
usually takes when the proposed equations involve incompatible 
conditions.” If, however, subjects of absurdity are not to be ab- 
surdly treated, I apprehend it will not require any extraordinary 
degree of reflection to be convinced of the incorrectness of such 
