146 



ME. T. T. WILKINSON ON 



(Fig. 3.) in a line M N of any order, such, that drawing the 

 tangents P V, P T, to two given circles (A) and (B), they 

 may have a given ratio;" but he does not particularise the 

 interesting cases just noticed, nor do I find in any of the 

 writings of this distinguished geometer* the least intima- 

 tion of his being aware of what the Continental Geometers 

 had effected by means of Radical Axes and Circles of 

 Similitude. 



When two circles (A), (B), and the tangential ratio are 

 given, he determines the circles of tangential ratio very 

 elegantly by taking « rad. AT:-1-HI::-LAD:BH, 

 in the given ratio, and drawing (to the circle through H, 

 centre B), the tangents DV*, D T', meeting AB in K 

 and L: — then KL is the diameter of the circle required," 

 (MSS., vol. vii., page G.) which construction, when the ratio 

 becomes that of AT : B V, reduces to the simple process of 

 describing a circle on the diameter of similitude, determined 

 by the direct and inverse common tangents to the two given 

 circles. (Davies, Diary, 1849, Prop. I.) The intersection of 

 the circle thus determined with the line M N, of any order, 

 furnishes a solution of the problem proposed by Mr. Swale. 



If we now suppose the circle (A), the circle of tangential 

 ratio (Q), the ratio of the tangents m : «, to be given, and 

 it were required to determine the other circle (B), we have 

 obviously the converse of the preceding general locus, which 



* An account of the peeoliar characteristics of Mr. Swale's geometrical 

 writings lias been given by Professor Davies, in No. VII. of his " Geomstry and 

 Geoinetergy'' in the Phil. Magazine for June, 1851 ; and a description of his MSS. 

 will be found in my "Additions to the late T. 8. Davies's Geometry and GeoToeiers" 

 in the Phil. Magazine for July and September, 1862. A short memoir of his 

 life may also be seen in the Mechanics'' Magazine for March, 1852. Mr. Swale, 

 although so long resident in Liverpool, belonged to the Yorkshire School of 

 Geometers, and was one of the ablest of Mr. Ryley's pupils. His able townsman, 

 the late Colin Campbell, Esq., also deserves honoiu-able mention amongst the 

 geometers of his time, but his elegant " Lucvhradons in Mathematics" do not 

 come within our scope, since their author was a native of Westmoreland, and 

 received his education ander Mr. Howard, of Newcastle-upon-Tyne. 



