12 Colorado College Studies. 



that which at the same time represents them all, viz.. the 

 equation x^ + y' = r' . 



The new equations show that all the comitants are 

 cubics; and the existence, as previously ascertained, of an 

 oval branch in each, at once refers them to that one of the 

 genera of Cayley and Salmon which consists of the projec- 

 tions of the '' par ahold cum ovali.^'' If a more minute 

 identification is desired, a transformation of coordinates, 

 easily effected, throws either equation into the standard 

 form discussed by Sir Isaac Newton in his '^Enumeraiio 

 Linearum Tertii Ordinis.'''' The curves thus prove to be of 

 the kind described by him as "defective hyperbolas having 

 a diameter,." and to belong to the species numbered 40, 

 whose distinguishing mark is the presence of the oval on 

 the concave side of the conchoidal branch. To write either 

 equation in the form adopted by Plticker as a standard, 

 no change of axes is necessary, but the equation of the 

 ir-comitants, for instance, becomes by a purely algebraic 

 modification, 



In this expression, the existence of the asymptote 



(?/— 2^=0) and of the "asymptotic point"' (0>-) becomes 



manifest, as also the position of the "satellite line" whose 



equation is (2 rlH — ]y=-r'^. Among the diametral de- 



\ 4/ 2 



fective hyperbolas — or, to use language better correspond- 

 ing to the vocabulary of Plticker, among cubics whose single 

 asymptote is osculating— the group numbered xxxiv is 

 distinguished by the fact that that line passes through the 

 asymptotic point, while the next preceding group, xxxiii, 

 differs from all others in which the asymptote and asymp- 

 totic point are separate, by the fact that the satellite line 

 does not cut the curve. In the special case in which I has 



the value -, the comitant belongs to xxxiv, otherwise to 



4 

 XXXIII. In either case, the existence and position of the 



