The Circular Locus. 21 



of each series may be known. Following the analogy 

 of the previous part of the problem, we may inquire 

 for the point, both of whose coordinates are pure im- 

 aginaries. Writing /I for x and ifj for y, the equation 

 becomes (m + /,a) u + (« + ?'>) /r^ — r-=0 and is resolved into 

 — /^> — •-"/ — r'=0 and wl-f;/r^=0. Now the point whose 

 coordinates are i^ and itj is the same as that whose 

 real coordinates are x= — >j, y=^', hence the foregoing 

 resolved equation may be interpreted as directing to 

 a point at the intersection of two lines, one of w^hich 

 is the polar of the known point v, —jj., and the other is 

 drawn through the origin at an inclination to the real axis 



whose tangent is -. The perpendicular distance of the 



point thus found from the real axis is in the same ratio to 

 its distance from the ,r-comitant as the unit of linear 

 measure is to the distance between the .r-comitants and 1; 

 and a similar proportion gives the interval between suc- 

 cessive 7/-comitants. 



Since every linear equation (unless its absolute term is 

 zero) may be thrown into the form MX + NY=r-, by multi- 

 plying it through by the (real or imaginary) ratio of ?•■ to 

 this absolute term, we have in the foregoing a direction 

 for constructing all the comitants of any such linear locus 

 given by its equation. The coefficients M and N, found in 

 this way, are the coordinates of the pole of the given locus. 



The problem of finding the points of contact of the two 

 linear tangent loci which have a given imaginary point in 

 common is equivalent to that of finding the points of in- 

 tersection of the circular locus by the polar of the given 

 point; and this may be done as follows, without the labor 

 of constructing the polar locus. The method is based on 

 an inspection of the result of a solution of the two equa- 

 tions xHY'=r' and MX-f ny = /-'. It is found algebraically 



that x-|-i'Y = ^(r±v/r-'— m'— n'). To express this re- 



M — IN *^ 



suit geometrically, we have first to find v/m^ + n*, a mean 

 proportional between m + Zn and m — /n; then'^/;•-— m^— n*, 



