i6o PETER GUTHRIE TAIT 



" Have you considered how far some of the geometrical proofs are independent 

 of anything that is distinctively Quaternions, and depend only on the notion of 

 ** +jy + kz, with i, j, k as incommensurable imaginaries not further defined ? " 



It was not till the summer of 1889 that the third edition began to 

 be printed ; and this naturally led to a renewal of the correspondence on 

 quaternionic subjects. Writing on June 15, 1889, Tait drew Cayley's 

 attention to a new problem which had been interesting him. 



In looking over the Trans. R.S.E. for your notes for the Fortschritte d. Math. 

 I suppose you saw Plarr's paper on the form of the spots which a blackened ellipsoid 

 would make if it were made to slide about in the corner of the ceiling. 



I have been trying to simplify the analysis, and have reduced the question to 

 one of mere elimination : but it is still very complex. 



With the view of studying what any point of the ellipsoid does, I had a very 

 true ellipse cut out of thick sheet brass in my laboratory, and have traced the curves 

 (of the 1 2th degree?) made by a pencil passed through various holes in it when it slides 

 between two perpendicular guide-edges. 



This was the beginning of Tail's discussion of the glissettes of the 

 ellipse and hyperbola. In reference to the problem Cayley remarked : 



" I abstracted Plarr's paper, but it did not seem to me that he had got out much 

 of a result not that I saw my way to doing it better. It is a very good question, 

 and a very difficult one. The plane question ought to be much easier tho' I fancy 

 even that might be bad enough. I shall be very glad to see your curves." 



On November 21, 1889, Tait referring to the glissettes wrote: 



Connected with the curves I sent you in summer there is a very curious theorem 

 which may, perhaps, be new to you. They can be traced by a point in the plane of a 

 hyperbola which slides between rectangular axes. 



A month later, Dec. 21, Tait wrote: 



My dear Cayley 



Thanks for your second splendid volume, which has come just in time for 

 my brief vacation, and contains in an accessible form the Quantics, which I have 

 long wished to read properly. 



The same post brought me a specimen copy of Quaternions, with various colours 

 of cloth to choose from. Brick red seems to be the most taking bait, so when you 

 get it you will have something striking to look at if not into. 



You will see, in a few days, in the Phil. Mag. another plea for Quaternions 

 as the physical calculus, par excellence. Perhaps it may lead to an increased sale 

 of my volume. 



Have you ever considered the locus of intersection of two normals to an ellipse 

 which are perpendicular to one another? 



