The Circular Locus. 27 



To illustrate by an example this class of inquiries, sup- 

 pose that we desire to determine at what point a line, tan- 

 gent to one of the comitants, cuts the same comitant again. 

 Let the o'-comitant /^ be the curve in question, then we 

 know that its tangent at the point M, N {i. e., the point 

 m+iti, n-\-i''', when these quantities satisfy the equation 

 M'-|-N"=7''), is the a;-comitant ii. of the locus MX + NY=r. 

 But the point at which these two lines again intersect will 

 not have the same coordinates, regarded as a point of the 

 linear locus, which it has as a point of the circular locus. 

 It is easily shown that a coincidence in position of points ^ 

 on two like a*-comitants belonging to different loci, icithout 

 a concurrent intersecfion of the y-comiianis, implies only 

 that the real parts of the coordinates x for the coinciding 

 points diif er by the same amount as the coefficients of i in the 

 imaginary parts of their coordinates y. If we represent 

 this unknown difference by d, we have the three equations 



( X + i!J. )- +{y + ir, Y=r\ ( m + ifj. y+{n + u )■= ?•' 

 and 



{x + ifj. + d) (m + i,".) + {y + ir,-\-id) (n-hiy) = r- 



from which to find x, y, and ry, eliminating d. Since each 

 of the equations can be resolved into two, by separating 

 the real and imaginary terms, we can eliminate also two of 

 the constants, as 7n and v, and find y in terms of r, /j., and n. 

 Two values of y will of course be found equal to n, and 

 represent the coincident intersections at the point of con- 

 tact, but the third, which is the subject of the inquiry, is 

 determined by the linear equation 



[(//.-— ?r)'+r7/'' ] y-j-fj. [ (/r— n-)'-f rXM'"'— 2 /r) ]=0. 



From this value of y, that of x and rj can be found, and the 

 tangential point is completely determined. As the equa- 

 tion is linear, no trouble arises from imaginaries. 



But let it be now required to find on the a;-comitant /j- 

 a point of inflection, using for the purpose the property 

 that at such a point the above-determined tangential point 

 must coincide with the point of contact. Then n is to be 

 substituted for y in the above equation, and the whole 



