8 Colorado College Studies. 



unit. Thus A=(/ + '«, B=6 + //^ etc. The Greek letter an- 

 swering to c is taken to be /', and that answering to y to 

 be >;. 



To return to the comitants, it is plain that one of each 

 series is of chief importance, namely, the comitant-zero. 

 This may be compared to the equator on a map, while the 

 comitant +1 and the comitant — 1 are on opposite sides like 

 parallels of latitude. The aj-comitant passes through all 

 the points for which X is real, and the ?/-comitant through 

 all those for which y is real, so that all "real points" of 

 the locus — that is, those having both coordinates real — 

 must lie on both these lines. For all real curves, there- 

 fore, the comitants zero will have a common arc, and this 

 arc will be — or, at least, will include — the curve itself, as 

 known to the ordinary constructions of analytical geom- 

 etry. Thus for the equation of our present example, 

 x*-|- y'= ?•", the it'-comitant consists of the circle of radius r 

 described about the origin, and in addition of a line coin- 

 ciding with the real axis. The ?/-comitant consists of 

 the same circle with the addition of the axis of imagina- 

 ries.* The other comitants, of either series, are cubic 

 curves situated on opposite sides of the axis named, and 

 having oval branches within the circle, as will shortly be 

 seen. 



A rule for constructing the comitants by points may be 

 derived by inspection of the equation. For the ic-comi- 

 tants, we may i3ut the latter in the form — y^=x^— 7-^, when 

 we have, by extraction of the square root, and the addition 

 of X to each member, 



X -f /y = x± \/(x + r) (X— )•). 



The mean propovtiomd between x-f r and x — r, which, 

 as the radical indicates, is to be found, is to be understood 

 of course not merely as having the length implied, in the 

 Euclidean use of the word, but as bisecting the angle at 



*Besido having the circle in common, these two comitants zero intersect 

 also at the origin, for this point of tlie locus may be regarded as having either of 

 its co-ordinates real, but since the latter cannot both be real at once, the origin 

 is not included iu the real part of the locus. 



