Prof. Young on the Theory of Vanishing Fractions. 517 
principle in question, as may be seen by a reference to the 
Algebra of Lacroix, where the circumstances of the problem 
are discussed at Jength. The ordinary expression for the ra- 
dius of curvature of a plane curve, will also furnish other ex- 
amples of the fallacy of the assumed principle; for when, in 
any particular example, that expression takes the form of a 
fraction, as r = ray we have, by differentiating, 
pean ee 
and it is well known that whatever values of x and y render 
this expression equal to zero, the same values, provided they 
fulfill the original condition, or equation of the curve, will be- 
long to points in it of maximum or minimum curvature ; or 
to points at which the contact with the osculating circle is 
above the second order. Now it is plain that the conditions 
P= 0, Q=O eressecscccceee’ (2) 
will cause a value of (1) to be zero; if, therefore, these con- 
ditions furnish for x and y values which satisfy the equation 
of the curve, the points to which they refer will be distin- 
guished from the other points by the order of contact being 
higher there than elsewhere. Instead of deducing this con- 
clusion from the expression r = 9? We Ought, in accordance 
with Mr. Woolhouse’s third principle, to say that at every 
such point the radii of curvature are innumerable, which is 
obviously absurd. As an example, let us take the common 
parabola, of which the equation is y? = 4m 2. By the usual 
process we obtain for 7 the expression 
_ fm+e = ALS 
8 x "4m? Q”’ 
and the conditions (2) are, in this case, 
Liege ab 
(m +a)%y= 0, Amat = 0, 
which are satisfied by the values x = 0, y = 0; and these 
values, fulfilling the original condition y? = 4m x, it follows 
that the origin of the axes, that is the vertex of the parabola, 
is a point at which the contact is above the second order, and 
this we know to be the case from other considerations, 
It is unnecessary to multiply examples illustrative of the 
fallacy of this third principle “as a general rule,” and indeed 
a passage in the reply of my Menta friend leads me to sus- 
pect that, while writing that reply, he himself had some mis- 
