The Circulak Locus. 28 



the point of intersection is now to be found as a third pro- 

 portional to =tL, c and r (that is, the triangle ROX is to 

 be made similar to LOG), and similarly Y is a third pro- 

 portional to L, s and r. (Figure omitted.) The quantity 

 x-f <'y might, of course, be found at once, as in previous 

 cases; but the construction of the separate coordinates 

 may be more useful.* 



It has now been shown how to find the intersection of 

 the circular locus by any linear locus whatever; a brief 

 mention should be made, however, of another mode of 

 attacking this problem, which may be best presented by 

 considering first the simpler one, to find the point of inter- 

 section of two given linear loci. Suppose that, in each, 

 the two comitants zero are given in position, also in each, 

 the o'-comitant I (the I having the same value in one as in 

 the other), and in each, the ^-comitant ry. Join the inter- 

 section of the ic-comitants to that of the j?-comitants c. 

 Now, since in either locus the distances from the d:;-comi- 

 tant to any two a;-coraitants, I and V , are as the numbers 

 ? and ^', it follows that the line just drawn will pass 

 through the intersections of any two like aj-comitants of 

 the two loci. So also the line joining the intersection of 

 the two ?/-comitants with that of the //-comitants r, will 

 pass through the intersections of all pairs of like //-comi- 

 tants. The i)oint in which these two lines meet is obvi- 

 ously the required point of intersection of the given linear 

 loci. 



The points of intersection of any two algebraic loci 

 whatever may be determined by an application of the 

 same principle; that is, the curves which pass respec- 

 tively through the intersections of like ^--comitants, and 

 through those of like //-comitants are always algebraic 

 curves, whose equations result from the elimination of ? 

 or of ri from a pair of comitant equations; and their real 

 intersections must always correspond in position with the 

 (real or imaginary) intersections of the two given loci. 

 But the application of this method to determine the inter- 



*Tho discussion of the intersections made witli a circular locus by imaRinary 

 radii leads directly to the tlicory of the trigonometric functions of an imaginary 

 variable, but this subject is much too extensive to be here entered upon. 



