148 



MB. T. T. WILKINSON ON 



(2.) When the circles are concentric and the sum of 



the tangents is constant. (Ques. 205.) 

 (3.) When the circles are any how posited and the sum 

 of the tangents is constant. (Ques. 212.) 



The Loci in cases (1) and (2) are shewn to be circles 

 concentric with the given ones, and the locus of (3) is an 

 ellipse; — when in (2) the sum of the squares of the tangents 

 is constant, Dr. Harvey and Mr. Butterworth have proved 

 in the Boston Enquirer, Ques. 72, that the locus is still a 

 concentric circle ; but if the same change is made in the data 

 of (3), Mr. W. S. Eyres, of Liverpool, author of an Essay 

 on Geometrical Analysis, has shewn in Ques. ^, ibid, that 

 the locus then becomes a circle, and if the difference be 

 substituted for the sum, we have then the locus a right line 

 which coincides with the radical axis when P T' — P V* 

 = O, and with the bisectant axis, f Diary, 1853, p. 79.) when 

 p T» — P V" == 2 ( A T* — B V*) = 2 (R' — r'), 



Mr. John Kay, of Royton, near Oldham, was one of the 

 most distinguished of Mr. Butterworth's pupils. His ques- 

 tions and solutions in the Gentleman's Diary, the Enquirer, 

 the Leeds Correspondent, and the Mathematical Companion, 

 are not very numerous, but they are usually characterised by 

 peculiar elegance and originality. Nor did he confine himself 

 to geometrical speculations alone, for in the Companion and 

 elsewhere we find him as the proposer and solver of some inte- 

 resting problems in Statics and Dynamics. His chief strength, 

 however, lay in pure Geometry, and from the masterly 

 specimens he has left us of his skill in the higher branches 

 of the Ancient Analysis, it would seem that he had not 

 failed to profit by the instructions of his able tutor. Several 

 curious properties of the Circle and the Conic Sections follow 

 from his investigationSj and in the difiicult subject of Porisms 

 he had but few competitors. The pages of the Mathematical 

 Companion contain the majority of his contributions to this 

 interesting branch of geometrical research ; — his solution of 



