Mr. Woolhouse’s Theory of Vanishing Fractions. 297 
reflect that 2, occurring in an analytical result, was as likely 
to be the symbol of absurdity, that is, of no value at all sub- 
sisting under the proposed conditions, as the symbol of mul- 
tiple values. 
When we are operating with equations of the first degree, 
containing several unknown quantities, the symbol $ is, in fact, 
the very form which the result usually takes when the pro- 
posed equations involve incompatible conditions; so that the 
foregoing theory would lead us to infer an unlimited variety 
of values, when in reality not one exists. ‘The theory which 
Mr. Woolhouse condemns could never lead to such absurdity. 
But even the examples which Mr. Woolhouse adduces do not 
appear to accord with the doctrine which they are intended 
to illustrate and enforce; nor do they furnish any ground of 
objection to the theory they are designed to oppose. Of these 
two examples the following is the one upon which, I believe, 
Mr. Woolhouse places the most importance. 
‘¢ To find a point in the arc of an elliptic quadrant, such 
that, a tangent being drawn through it, the perpendicular 
drawn from the centre to the tangent may be a mean propor- 
tional between the two semiaxes a,b.” Now by putting x, y 
for the coordinates of the required point, we easily obtain the 
following equations, embodying the proposed conditions, viz. 
Eyed y ae 1 x y 
JERE cabs Tresabe hn ae 
¢¢ and we find 
so that the coordinates of the required point are 
iN SH {MOTE bs, 
e=ay/p y= NY fs 
Now, although most persons would say that these results 
furnish al/ the values of x and y legitimately deducible from 
the preceding expressions for z* and y*, yet Mr. Woolhouse 
adds, “* When 6 = a the elliptic quadrant becomes a circular 
one, and these last expressions give for the position of the re- 
quired point x = a 3, y = a 3, or the point which bi- 
sects the arc of the quadrant. But in the case of the circle, 
it is obvious that all its points will answer the proposed con- 
Third Series. Vol. 8. No. 47. April 1836. aI 
