Prof. Young on Conjugate Points. 175 



Our proposition is therefore proved; and all the conclu- 

 sions that follow from the contrary doctrine are at once de- 

 stroyed. In truth, could we for one moment expect this con- 

 tradiction between the parts of the hypothesis to result in any- 

 thing else than the symbol of impossibility? Might it not, in- 

 deed, be more properly called the symbol of contradiction! 



Of course with the fundamental "assumption," as Mr. 

 Warner very truly calls it, all that is built upon it falls; and 

 it might seem almost unnecessary to say another word on the 

 subject. However, with respect to the conjugate hyperbolas 

 which are so often quoted in discussions of this kind, it may 

 be worth while to remark, that they cannot be brought under 

 the same equation 'with the primary ones. Neither (though this 

 is generally lost sight of) do they belong to the same geometri- 

 cal system. They cannot be cut from a pair of opposite right 

 cones conjugate to those from which the primary ones were 

 cut. They are, in fact, the produce of a second geometrical 

 hypothesis as much as their equation is. * 



Points out of the co-ordinate plane X Y are as foreign to 

 the original hypothesis respecting a plane curve, as the point 

 C is foreign to that of A B = +a and A B'= — a. In short, 

 the entire speculation is unworthy of the ability which has 

 been employed upon it; and its prosecution throws discredit 

 on the scientific honours which many of its cultivators have 

 deservedly won. 



August b, 1846. Shadow. 



XXX. Note on Mr. Warner's Paper on Conjugate Points. 

 By J. R. Young, Professor of Mathematics Belfast College. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



[ THINK with you, that Mr. Warner's views on the sub- 

 •*■ ject of conjugate points — as delivered in the last Number 

 of the Philosophical Magazine — are not likely to meet with 

 acceptance among mathematicians; and I am persuaded that 

 even Mr. Warner himself will feel inclined to abandon them 

 when he reflects upon the consequences to which they lead. 

 His fundamental analytical principle is, that " +0= —0;" 

 from which it follows, taking the reciprocal, that + oo =s — oo; 

 and therefore, transposing, that 2» = 0, that is, that a quan- 

 tity infinitely great is nevertheless equal to nothing. His 

 fundamental geometrical principle is of like character : it is, 

 that a finite and determinate line (D E), merely by being 

 turned about a little, suddenly becomes infinite. Mr. War- 



