The Circular Locus. 11 



infinity, also all the ovals are mutually tangent at the 

 origin. Beginning- with a very small value of //, the corre- 

 sponding comitant of course lies near the comitant zero, 

 which consists of a circle and a line; and it is seen that 

 the conehoidal jjart of the comitant answers to one semi- 

 circumference and to that part of the line which is outside 

 the circle, and lies just outside these parts of the comitant 

 zero. The oval, on the other hand, answers to the other 

 half the circumference and to its diameter, and lies inside 

 the semicircle close against the boundary. Accordingly, 

 its curvature is rapid on the side remote from the origin, 

 while near the latter it is almost straight. The succeeding 

 comitants have their conehoidal parts successively further 

 and further from the axis, each encomiaassing its predeces- 

 sor, and more nearly straight than it, while the ovals lie 

 each one within the preceding, and are rounder as well as 

 smaller. Hence for /^==oc, the comitant should consist of 

 a point (the origin), and a line at an infinite distance. 



We are thus able to construct and describe the systems 

 of comitants from the equation x^ + y"^=?-' alone; but if 

 we wish to determine the order of these curves, so as to 

 study them in the light of related forms, it becomes de- 

 sirable to pass to new equations. In the article already 

 mentioned, contributed to the Annals of Mathematics, it is 

 shown that from the equation of any locus we may derive 

 that of another, whose real part shall coincide with any 

 specified comitant curve of the given locus. In the present 

 instance we shall obtain, for the general ^r-comitant — the 

 ic-comitant c — the following result: 



x^y + if —r'i/—2:{x'-i-y-)=0[ 



and for the 7/-comitant /;, 



J'" + Jcy- — 1 -'.r -I- 2 // ( jr -f ^' ) = 0. 



These may be called the "equations of the comitants," for 

 the sake of brevity of expression, if it be borne in mind 

 that from our present point of view the only equation 

 which truly represents the comitants and nothing more, is 



