22 Colorado College Studies. 



a mean proportional between j'+v^M^ + N^and r— •v^M''H-n"', 

 and finally a fourth proportional to m — ?'n, r, and 



If P is the given point, whose coordinates are M, n, and 

 if M is the point m, 0, then MP is equal to /n, and by pro- 

 ducing PM to Q. making MQ=PM, the point m, — n, is 

 found. Hence the first step is accomplished when a dis- 

 tance equal in magnitude to a mean proportional between 

 OP and OQ is laid off both ways from O on the line bisect- 

 ing the angle POQ. From each of the points thus fixed a 

 distance r is then to be measured to the right, parallel to 

 the real axis, so fixing the points H, K. Now the angle 

 HOK is to be treated as was POQ; ?. e., on its bisector is 

 to be laid ofiP in each direction from O a length equal to a 

 mean proportional betw^een its sides, and from each of the 

 points so found a distance r measured to the right, to the 

 points E and F respectively. If K be the right-hand ex- 

 tremity of that diameter of the circle which lies on the 

 real axis, we have next to make a triangle OET similar to 

 OQR, on the base OE homologous to OQ (and with the 

 angle TOE equal in sense as well as in magnitude to ROQ), 

 and Twill be one of the required points of contact; the 

 other having the same relation to OF that T has to OE. 

 (Figure omitted.) 



The remark has been already made that any linear 

 locus which does not contain the point 0, 0, may be re- 

 garded as represented by the equation MX-f ny=?-*; hence 

 the foregoing construction of the intersections of such loci 

 needs only to be supplemented by discussing those of the 

 locus CY=sx, to complete the treatment of linear intersec- 

 tions. Let the points C and S be the extremities of vectors 

 extending from O and having the values c and s respec- 

 tively, and let P be the point whose (imaginary) coordi- 

 nates are c, s. Produce PC to Q making CQ = PC, then the 

 vectors OP, OQ are c + is and c— /s respectively; so that 

 OL and OL' (denoted by dzL) will represent ±v/c' + s', 

 if L and L ' be taken on the bisector of the angle POQ and 

 at distances " each way from O equal to the mean propor- 

 tional between the lengths OP, OQ. The coordinate x of 



