ANALYSIS OF CORRESPONDENCE 141 



the dates of Oct. 25 and Dec. 2, 1859. The discussion is reproduced in 

 essence in the Elements, although much more briefly. 



(2) Transformations connected with Fresnel's wave-surface. Tait 

 began the discussion in letter 10 and continued it in many of the subsequent 

 letters down to letter 34. Hamilton took up the theme in letter xiv and 

 elaborated it in letters xv, xv', xv", which ran on consecutively for 

 96 pages. Here also the essential parts of the investigations both of 

 Hamilton and Tait will be found in their works. In letter 20 Tait suggested 



the use of the form <j> and in letter 23 gave the wave-surface equation in 



the new form T (p* + <f?)~ p = i ; a. form whose elegance Hamilton at once 

 recognised and continued thereafter to use. 



(3) The theory of the linear vector function. This is chiefly contained 

 in Hamilton's xix, xxiu, xxv, and in Tail's 32 and 33. The essential parts 

 are reproduced in Hamilton's Elements and in Tail's Elementary Treatise. 



(4) The theory of envelopes. This was begun by Tail's problem 

 of the paraboloid cylinder which forms section 321 of his Treatise (3rd edilion). 

 The problem greally look Hamillon's fancy. He began ihe discussion in 

 letter vm, and developed it in elaborate detail by quaternion processes in 

 letters xi and xm. 



(5) The planning of the new ireatises on the calculus. Early in 1859 

 Hamilton began to write his " Manual," which finally appeared in 1866 as the 

 Elements, unfortunately incomplete in consequence of ihe dealh of the author 

 in 1865. Tail's own trealise was projected during the summer of 1859, bul 

 was wilhheld from publicalion until Hamillon's work should appear. It was 

 finally published in 1867. 



In connection with the preparation of Tail's Quaternions the following 

 letter to Sir John Herschel is of considerable interest. Tait had senl Herschel 

 copies of some of his qualernion contributions to ihe Quarterly Journal of 

 Mathematics and, in reply lo Herschel's acknowledgement, wrote on Dec. 14, 

 1 864, as follows : 



My Dear Sir 



I am much obliged by your very kind note just received.... 



Five years ago, Messrs Macmillan & Co. advertized for speedy publication 

 an " Elementary Treatise on Quaternions " by me ; but, as my good friend 

 Sir W. R. Hamilton thought that it might possibly interfere with his forthcoming 

 " Elements of Quaternions " I withdrew it and have published only the few articles 

 I recently sent you all of them with his approval. 



I had no idea that you had been engaged in preparing such a work ; and I 



