CORRESPONDENCE WITH CAYLEY 155 



disregarding the scalar part w, as the force represented by the lines x, y, z\ and in 

 the second form, disregarding the tensor r/sin \f, as a rotation f about the axis 



(, ft, 7). 



Then sum of two quaternions, qua force, is the resultant force. 



Product of two quaternions, qua rotation, is the resultant rotation. 



But is there any interpretation for the sum qua rotation or for the product qua 

 force} It would be very nice if there were. 



We enjoyed our American expedition very much. I was glad to hear from 

 Thomson that he also was going to lecture at Johns Hopkins University.... 



Yours very sincerely 



A. CAYLEY. 

 CAMBRIDGE, 



yd Nov. 1882. 



UNIVERSITY OF EDINBURGH, 



4/11/82. 

 My dear Cayley 



I was very glad to get your note, and to hear that you had enjoyed 

 your venturous journey. Thomson's proposal was quite new to mel I have not seen 

 him for months. 



I am absolutely overwhelmed with work just now ; as, besides my University 

 work, and R.S.E. do., I have been virtually forced to give a course of lectures to 

 ladies, and I am writing, against time, a very long article for the Encyc. Brit. 



Maxwell's death left the staff of the Encyc. in a state of great perplexity. He 

 had drawn up a scheme for the scientific articles, and had done the greater part of 

 the work himself. Had he lived, the article " Mechanics " would have been written by 

 him, or entrusted to some competent writer, two years ago, at least. As it is, the 

 acting editor discovered, only three months ago, how much had been referred forward 

 to it; and I spent the greater part of my summer holiday in writing it Seeing 

 it through the press is no joke! And the work of trying to boil down the whole 

 of abstract dynamics into 60 pages has been very heavy. 



I fear I misunderstand your questions. Of course I know that Vq is a force 

 and that V(q + r) = Vq + Vr; whatever quaternions q and r may be. But, as to 

 rotation, I have always written (after Hamilton) 



9( )r' 



where (of course) we need not trouble about the tensor. This gives qr( )r~ l q~ l 

 as the result of r( )r~ l followed by q( )q~ l ; and may be written 



qr( }(qr}-\ 



Now, in asking about the interpretation of a sum qua rotation, do you mean the 

 effect of (q + r) ( ) (q + r)~ l ? Also, as to the product qua force, do you refer to 

 V.qrl 



I can easily answer these questions, but I fear I have not caught your mean- 

 ing.... 



Before Cayley's reply to this was received, Tait wrote a second note. 



