18 Colorado College Studies. 



The rule that imaginary intersections of a circular and 

 a linear locus must be conjugate, and hence discrete, ceases 

 of course to apply if the linear locus is imaginary, when 

 two imaginary intersection^ may be coincident. In this 

 case, at the point of contact of the two loci an j--comitant 

 of one must be touched by the corresponding j--comitant 

 of the other, and also a ?/-comitant of the first by its cor- 

 responding ^-comitant of the second, the two curves hav- 

 ing in general different directions at the point, since in an 

 imaginary linear locus the two series of comitants need 

 not be parallel to each other. Two such linear loci, having 

 a real point Q in common, may touch the circular locus. 

 The usual form of this statement, that "two imaginary 

 tangents to a circle may be drawn through any point within 

 the latter," must not be allowed to give the impression 

 that the comitants of the linear loci, which are the actual 

 tangents, will pass through the point Q. This would of 

 course be impossible, since Q, as the sole real point of each 

 linear locus, is at the intersection of its comitants zero. 

 In fact, since the polar of Q with its comitants constitutes 

 a real linear locus, intersecting the circular locus in the 

 points of contact, the latter points must, as already shown, 

 be on the perpendicular from the origin on this polar; that 

 is, on the line OQ. ^ 



The complete geometric construction, exhibiting the 

 '' tangents to a circle from a point within," may be made 

 as follows: (See figure, page 19). 



The circle drawn around the origin as center, and the 

 point Q (which, to avoid undue simplification, may be 

 assumed to be on neither axis) are the only data needed. 

 The polar of Q is first to be drawn, and the intersections 

 made by the locus to which it belongs are to be found in 

 the manner just stated. These are the points of contact. 

 By a previous construction the coordinates of T, one of 

 these points, may be determined geometrically, and it 

 will be well (in order to render apparent the significance 

 of the subsequent construction) to sketch at the same 

 time the circular comitants which i3ass through T. Now 

 if the algebraic expressions for the coordinates of T be 



