Mr. H. S. Warner on Conjugate Points. 87 



Again, for x — 2a^ / 2 (#) = = + 0//— 1, 

 while for x = 2a + /i,[tis=—(a + h)\/k; 



and for 2 a — h, 



f % {x)=-{a-h)V-h=-{a-h).hh \f-l; 



or possible for 2a + h, and impossible for 2a — h; hence 

 there is a cusp determined by 



x — 2 a, y =/ t {2 a) + = 2 a. 



9. It will be seen that these rules are substantially the same 

 as those of Prof. Young ; the only difference is, that he says 

 that we must find a value of y in which the symbol Ov' — 1 

 appears, and then suppress this evanescent imaginary quan- 

 tity; while I say that it suppresses itself, being equal to 0. I 

 have considered only the imaginary V— 1, but it will be 

 found that the same principle applies to other imaginary ex- 

 pressions, such as log — 1, sin is/ — 1, sin -1 V' — 1, &c, 

 namely that each of these when multiplied by 0, is equal to 

 0; the reason being that no quantity but oo can, when mul- 

 tiplied by 0, produce anything but 0, and these imaginaries 

 are not equal to oo . 



Perhaps in a future Number I may consider some other • 

 points connected with these subjects. 



Trinidad, January 3, 1846. 



Postscript. 



Port of Spain, Trinidad, 

 April 3, 1846. 



I find that I inadvertently made use of the term cusp in 

 speaking of that point at which a curve ends abruptly. This 

 certainly was an error, but I know not what name to give to 

 this point instead of cusp ; for there appears to be no recog- 

 nised appellation, as yet, in consequence of the little attention 

 that has been given to this peculiar kind of point. Neither 

 the "abrupt termination" of Prof. De Morgan, nor the 

 "point d 'arret " of the French writers seems to me to be a 

 good name for it. Query : why should we not call it a ter- 

 minus! We might then define a cusp as the common termi- 

 nus of two or more branches of a curve. 



H. S. Warner. 



