28 Colorado College Studies. 



solved for n. Oue root, »=//, ran be immediately found, 

 and this will be seen to indicate the existence of a point of 

 inflection at an infinite distance. After dividing out the 

 factor n—.'j; there remains the biquadratic 



n' + 2//?i'— 2// (//+?•-) 7?— /r(/r+?--)=0. 

 Two of the roots of this equation are imaginary, and with 

 them a third must be rejected on the ground that it cannot 

 consist with real values of m and v. The remaining root 

 shows the position of the two real, finite, points of inflec- 

 tion. The three rejected roots are just sufficient in num- 

 • ber to represent the six imaginary inflections of the cubic 

 curve, whose real part coincides with the ic-comitant ,«; 

 but it seems plain that, consistently with the fundamental 

 notions, of the present scheme of interpretation, it can 

 only be said that the latter curve, considered as a comitant 

 of the locus X"4-Y'=r", does not possess these six inflec- 

 tions. 



As all of the foregoing discussion has been limited to 

 the interpretation of the equation X"+Y'=r, the circular 

 locus which has been treated is not the most general one 

 to which the name applies, but is merely the real circular 

 locus having its center at the origin. It seems proper in 

 a closing paragraph or two, to indicate what modifications 

 of the foregoing constructions would result from their ex- 

 tension to the circular locus in general. 



The removal of the center from the origin is effected 

 by a parallel transformation of coordinates, in resjpect 

 to which it is only necessary to remark that transformation 

 to a real point simply changes the relative position of the 

 axes in respect to the entire system of curves, leaving the 

 latter unaffected, while transformation to an imaginary 

 point disturbs, not the form of the comitant curves, but 

 the numbers by which they are characterized, thus the 

 comitant zero may become the comitant p-, while some 

 other comitant of the same series becomes the new comi- 

 tant zero. From this statement, the effect of such a change 

 on the foregoing geometrical constructions may be easily 

 estimated. 



