THE LANCASHIBE GEOMETERS AND THEIB WRITINGS. 



14.5 



also by the proposer, but the former gentleman's solution is 

 the more remarkable on account of the following "curious 

 Theorems on Circular Loci" which he enunciates at the close 

 of his investigation : — 



(1.) If two lines be drawn to meet each other, one 



from a given point, and the other a tangent to 



a given circle, such, that they shall have a 



constant given ratio, but not that of equality, 



the locus of their points of intersection is a 



circle. 



(2.) If the tangents of two given circles be drawn 



to meet each other and have a constant given 



ratio, but not that of equality, the locus of their 



points of intersection is a circle. 



The first of these Theorems is obviously that case of the 



second when one of the circles is reduced to a point; — the 



locus in both cases is the " circle of tangential ratio," and if 



the given ratio becomes that of the radii of the two circles, 



we obtain the circle of similitude as a particular case of the 



general Theorem. (Davies, Diary, 1851, Prop. XII.) The 



case, when the given ratio is one of equality, is specially 



excepted by Mr. Nicholson, but it was subsequently proposed 



for an independent solution in the Companion, and also forms 



a corollary to Mr. Lowry's solution of (2) in Ques. 117 of the 



New Series of the Repository. In both the places cited the 



locus is shewn to be a right line, since termed the radical 



axis of the two circles, and is, in fact, that case of the general 



locus when the radius of the circle of similitude, or of the 



circle of tangential ratio, becomes of infinite magnitude. 



The publication of Lowry's Lemma and Corollary in the 

 Repository appears, by a reference, to have led Mr. Swale to 

 enter more fully into the subject of Tangential Ratio than 

 had been done by his predecessors. In his MS. remains, 

 (vol. vii., pp. A to L.) he constructs all the principal cases, 

 and extends the inquiry to that of determining " the point P, 



u 



