10 Colorado College Studies. 



radii of any convenient length, describe arcs meeting at F; 

 draw OF, and on it take OL and OL' in opposite direc- 

 tions, each equal to OQ.* On the side of LOL' toward X, 

 describe arcs from centers L and L', with radius OX; and 

 intersect each of them by an arc drawn from center X with 

 radius OL. Then P and P', the points of intersection, will 

 be points on the comitant curve, one on the infinite branch, 

 the other on the oval. Two more points on the same comi- 

 tant are situated symmetrically to these, on the opposite 

 side of the imaginary axis. Also, the ir-comitant — 'j. is 

 equal to the a;-comitant //, and the real axis is an axis of 

 symmetry to the two curves. Further, if either of them 

 be rotated through a right angle, a ?/-comitant is produced; 

 but it must be noted that while, for all positive values of 

 //, the infinite branch of the a:-comitant ,". is situated above 

 the real axis, that of the ^/-comitant !>■ is to the left of the 

 axis of imaginaries. Thus a single construction virtually 

 determines sixteen points — four on each of two £c-comi- 

 tants and as many more on two y/-comitants. 



Each comitant, excepting the comitants zero, proves 

 when drawn to consist of a conchoidal and an oval branch, 

 the former aj^proaching at infinity a line drawn parallel to 

 an axis (the real axis in the case of £c-comitants) and at a 

 distance therefrom equal to double the coefficient {it) of /, 

 which distinguishes the particular comitant; while the oval 

 lies on the opposite side of this axis, touching the latter at 

 the origin. The breadth of each branch, in a direction 

 perpendicular to the above-named axis, is the same, and 

 is easily found to be n/^-'^+.u^— //; while >/?-'^+,a^-f // is the 

 greatest ordinate of the curve, belonging to the point where 

 the conchoidal branch cuts the other axis, which divides 

 it symmetrically. Regarding together all the comitants of 

 one series, it is apparent that all have a common point at 



♦Another method of finding a geometric mean between two linos which ex- 

 tend from a common point is as follows : First, bisect the angle of the lines and 

 also its adjacent supplementary angle. On tlio latter bisectrix lay off half the 

 longer line, and from its extremity, in both directions, half the shorter, thus 

 obtaining the half sum and half difference of the lines. From the extremity of 

 the half-difference, witli radius equal to tlio half sum, cut off on the bisectrix of 

 the angle a segment, which will be the required mean proportional. 



