20 Colorado College Studies. 



tants of the other tangent locus may be drawn through T ' , 

 the second point of contact. 



By a slight modification of this construction, a tangent 

 may be drawn at a given imaginary point of the locus. 

 For if T be given, the points of the tangent locus for which 

 X and Y respectively vanish may be f(mnd as before, but Q 

 must be determined in order to obtain the directions of 

 the comitants zero. From the relations already stated it 

 is easily shown algebraically that if OT be a given dis- 

 tance d, then OQ is found from the proportion 



d'+r': r- =2 d lOQ; 

 which admits of ready geometric construction.* 



Again, the construction of the jjolar of an imaginary 

 point whose coordinates are given is obtained in a similar 

 manner. For the equation of the polar of a point M, N is 

 in the same form MX + NY = r- already used for the tangent, 

 hence the points of the required locus for which x and Y 

 are severally zero are found as before, but the real point 

 must be separately determined. For this purpose we may 

 put for M and N respectively their expanded values m-\-i'j; 

 n-j-V'' (where m, n, !>■ and v are given real quantities), and 

 then if we assume that x and y are the real quantities x 

 and y, we can separate the equation {m + i!>) x-\-{n-\-h)ii— 

 7--=0 into two (since the sums of the real and imaginary 

 terms must sejjarately vanish), which will be mx-\-ny—r'^=0 

 and iJ.x-\-'-'y=(i. The real point of the locus is then at the 

 intersection of two lines, whereof the former is the polar 

 of a known point m, n; and the latter is a line through the 

 origin, the tangent of whose inclination to the real axis is 



— — . With the determination of this point, two points of 



V 



each of the comitants zero become known, and hence these 

 lines may be drawn. To fully construct the locus, hoM-- 

 ever, it is needful to have the coordinates of one imaginary 

 point, that the distance between successive comitants 



*In this is included the solution of a problem relating to a comitant curve 

 considered as a variety of the cubic, and independently of its connection with 

 the other comitants forming the locus ; viz : To draw a tangent at a given point 

 of the curve. 



