30 Colorado College Studies. 



and must be classed among Newton's " defective hyperbolas 

 not having a diameter." The "comitant equations" are 

 3r-y+if—2^x--\-y-) + 2r(>.x+{/r—i'')y=0, 

 oc!" +xy- -{-2rj {x' -{- y") -\- {f>-—i-) x—27-p.y={); 

 (and from these as general forms the special cases already 

 considered are derived by making /> or r equal to zero.) 

 All the comitants of a series pass through the origin, and 

 have there a common tangent. This tangent, however, for 

 the a;-series, is no longer the real axis, but has a slope of 



2rp 



— ;,' SO that in the loci in which p = ±r, the common 



r"— co- 

 tangent coincides with the imaginary axis. The infinite 

 branches of the comitants are not conchoidal, but serpen- 

 tine, crossing their asymptote (which is still jDarallel to 

 the real axis, and at a distance 21 therefrom) at a point 



(t' — /0°) ^ 



distant by -'^ — from the imaginary axis. In the case 



^ rp ^ J 



of the comitant zero, this crossing is at the origin, and the 

 serpentine curve is symmetrical to that point as a center, 

 thus belonging to the Newtonian species numbered 88. But 

 as I increases from the value 0, the successive comitants, no 

 longer possessing any kind of symmetry, belong at first to 

 the species 37, as they have no oval and no singularity. 

 There is, however, the same gradual closing into the form 

 of a narrow-necked purse, already observed in the case 

 ?*=0, and the comitant p is again nodate, so belonging to 

 the 34th species. For still greater values of I, the curve is 

 of species 33, having an oval, which, as in all cases, shrinks 

 to a point at the origin as I becomes infinite, the serpentine 

 branch in the meantime straightening toward coincidence 

 with its asymptote. The node on the a;-comitant p is at 

 that point of the locus for which Y=0; and it may be re- 

 marked, as true of all forms of the locus, that the four 

 points characterized by zero values of one or the other co- 

 ordinate are double points of the comitants of opposite 

 name on which they fall. 



Very slight and obvious modifications are alone required 

 to adapt the first method given in the present paper for the 

 construction of comitants, to use with any given values of 

 r and p. 



