388 ON THE AXES OF A CONIC IN TRILINEARS. 



obvious, that it may be supposed Hamilton did not in such considera- 

 tions really intend to prove the veracity of consciousness, but rather 

 to illustrate the necessity of admitting it by indicating the conse- 

 quences implied in its denial. 



The length, to which this article has extended, has allowed me 

 merely to touch on this last subject. I am compelled also, for the 

 same reason, to omit any reference to the rules, laid down by Hamil- 

 ton, for applying the testimony of consciousness or to his claim of 

 originality in their discovery. 



ON THE AXES OF A CONIC IN TRILINEARS. 



In vol. IX., No. 52, and vol. X, No. 59 of this Journal, were given 

 some descriptions of particular cases of the trilinear equation to a conic. 

 In the corresponding discussion of the general equation, it does not 

 appear to have been noticed that the axes can be determined by the 

 very same process as that used by Sir W. Thomson to obtain the prin- 

 cipal radii of curvature at a point in a surface, the resulting equation 

 differing only in the forms of the constants. This omission has been 

 caused, I believe, by the systematic neglect of the relation (2) of this 

 article, which was demonstrated and employed in the articles above 

 referred to. I proceed to the general investigation. 



Let the conic 

 <^ (a, /3, y) = wa2 + vfi^ + wy^ + 2M'/3y + 2y'ya + 2iv'a^ = 0. 

 be cut by the diameter 



« — / _ ft — ff _ y - h _ ^_ 

 I m n ' 



where the point (/, ff, h) is the centre, ?• is the distance between 

 (a, /3, y) and {/, ff, h), and I, m, n, are subject to the conditions 



al + hm -\- en :=■ 0, (1) 



P sin2^ + m2sin25 + w^ sin 2 C= 2sin ^ sin^sin G. (2). 



Then the two values of r are given by the equation 

 </) {I, m, n) r^ + (f) (/, ff, Ji) — 0, 

 or, writing 



- P for 4> (/, g, h\ 

 P 

 — = uP + vm^ + ivn^ -\- 2u'mn + 2v'nl +2ivhi. 



^2 



