1831 .] 
363 
On Conjugate Hyperbolas . 
« 
the tangent of COP, or multiplied by the tangent of CON. These hyperbolas, 
therefore, cannot be conjugate, except in one case, that is, when the conjugate cones 
are both equal ; or, in other words, when the angles COP, CON are each 45°. In that 
ease r = 1 — and * = b. 
t t 
To get the true conjugate hyperbolas, the cones must be cut in a different man- 
ner. The planes of the hyperbolas, instead of being the same, must be at right angles 
to each other ; that is, in figure 2, pi. XIII. let CFA, CWE be the planes of 
the vertical triangles of the cones, and let YTX, YIZ be two planes parallel to 
them, cutting the cones at right angles to each other, and to the bases of the cones, 
viz. PAO, OZL, then the hyperbolas formed by the intersection of YTX with 
the cone PCO and YIZ with OCL, are conjugate to each other. 
The truth of this is obvious, since it is plain, that TF, the distance of the plane of 
the hyperbola YXT, from the plane of the vertical triangle CFA, is equal to the 
semi-transverse axis of the hyperbola YZI ; and similarly IW, the distance of the 
plane of the hyperbola YZI, from the plane of the vertical triangle CWE, is equal 
to the semi-transverse axis of the hyperbola YXT. It follows, therefore, that to pro- 
duce opposite hyperbolas, the cones must be cut by the same plane in the manner 
commonly directed; but to produce conjugate hyperbolas, the conjugate cones must 
be cut by planes perpendicular to each other. 
I shall forbear considering the second case, that in which the planes of the hy- 
perbolae are inclined to the plane of the base, as it would extend this letter too much ; 
and my chief object is attained by having pointed out the error of the common 
idea of Conjugate Hyperbolas. I may, perhaps, address you on it at a future time. 
In Rees’ Encyclopaedia, art. Hyperbola, it is stated. “ It may be remarked, that 
while opposite hyperbolas must be regarded as two different branches of the same 
curve, conjugate hyperbolas are two different curves, possessing, indeed, some ana- 
logous properties, but really unconnected by the law of continuity. For in the first 
place, when a plane cuts two opposite conic surfaces, it produces no more than two 
opposite hyperbolas, without the smallest trace of the conjugate hyperbola; and in 
the next place, if we consider the hyperbola as it is determined by an algebraic 
equation, no such equation can be found, that preserving the same system of 
the co-ordinates, will comprehend all the four conjugate hyperbolas.” 
It may be observed, that although the hyperbolae so cut from conjugate cones in 
the common way, are not conjugate to each, yet they are similar to these conjugates. 
This is noticed in the new edition of Hutton’s Course of Mathematics, edited by 
Olinthus Gregory, 1828, of which a few copies are just come out to India, and which 
contains many valuable additions to the original work. In Vol. II. page 105, the 
last sentence of the quotation, which I have already given from the old edition, is 
altered to the following : “ There will be formed the four hyperbolas GQR, FST, 
VKL, WHI, of which each two opposites are equal ; and each pair resembles the 
conjugates to the other two, as here in the annexed figure; but they are not accu- 
rately the conjugates, except only when the four cones are all equal; and then, the 
hyperbolic sections, are all equal also.” No directions, however, are given how to 
cut the accurate conjugates. It may also be remarked, that the expression, “ each 
pair resembles the conjugates of the other two,” is not correct mathematical 
language. It should be “ each pair is similar to the conjugates of the other 
^o,” as may easily be proved ; for they are between the same asymptotes, 
and hyperbolas that have the same asymptotes, are similar to each other. 
This may also be shewn thus ; the transverse axis of the hyperbola cut from 
