2G Colorado College Studies. 



section of the line x cos ct + y sin ^=p, when j; was made 

 infinite, was found to be infinitely distant on the radius 

 boundin<ij the an,u;le (f. 



Thus the circular points at infinity are not definitely 

 localized, but it is to be noted that their indeterminate- 

 ness results, not from their imaginary quality alone, nor 

 yet wholly from the infinity of their coordinates, but from 

 the combination of both characters, and the absence of 

 any means of imposing a definite value upon the ratio of 

 the real and imaginary parts. 



The problems which have been discussed in the present 

 paper are mostly of the class which relate to properties 

 of the locus considered as a whole, rather than to those 

 of individual comitants. As such problems when analyzed 

 will, as a rule, resolve themselves into investigations of the 

 intersections of the locus by other loci, they will generally 

 be distinguished by the mark that the comitants of the 

 two series enter similarly into the geometrical construc- 

 tions; thus at the intersection of two loci the like a?-comi- 

 tants and also the like ^/-comitants meet in the same point. 

 Of the other class of investigations mentioned, viz., that in 

 which the properties of individual comitants are the subject 

 of inquiry, an instance occurs in obtaining the " comitant 

 equations." In such problems it is frequently necessary to 

 employ separate symbols for the real and imaginary parts 

 of constants or of variables; and if in the subsequent 

 course of the investigation, a symbol which has thus been 

 taken to represent a real quantity occurs as the representa- 

 tive of the unknown quantity in an equation, imaginary 

 roots of such an equation must of necessity be treated as 

 impossible, since they contradict the hyi^othesis of reality. 

 This is just as if m, having been put to represent the in- 

 tegral part of a mixed number, in some algebraic problem, 

 should afterward be determined by a quadratic equation, 

 one of whose roots should prove to be integral, the other 

 fractional. The latter root would of course be disregarded, 

 without the imputation of any unreal quality as belonging 

 to fractions. 



