THE WAVE SURFACE 125 



Hamilton acknowledged the receipt of this letter by sending the first 

 instalment of letter xiv. 



OBSERVATORY, Nov. \Tth. 1858. 

 My dear Mr Tait 



Although X and XIII are still unfinished, not to mention IX, which 

 is little more than begun, I am in a mood to commence now a new letter, of a 

 perfectly miscellaneous nature, and free from the tyranny of any fixed idea. 



You tell me that you have been making progress with treatment of Fresnel's 

 wave by Quaternions, but that you have not (or had not at the time of writing) 

 completed the investigation. Whenever you have quite satisfied yourself with a 

 result, or set of results, upon that subject, I should prefer you not immediately 

 communicating such result, or results, to me ; because I should like to try, either to re- 

 investigate the equation of the wave, or perhaps to hunt out an old investigation of 

 it, in one of my manuscript books. The fairest, or at least the pleasantest course 

 for both of us may therefore be, that we should agree upon some day and each of us 

 on that day post a letter containing some of our separate results. 



This suggestion was warmly welcomed by Tait ; and in his letter 1 2 of 

 date Nov. 29, 1858, the following reference was made to the agreement : 



You mentioned no day in particular for our exchanging results on the Wave 

 Surface. I have (in a sense) completed my investigations but they are far from 

 simple and I suspect strongly that there is some very elementary theorem of Trans- 

 formation with which I am not acquainted which would immensely simplify them 

 at once. I would therefore, to avoid knocking my head longer against eliminations 

 which at present I find impracticable though I know they must be possible, request 

 you to name as early a day as may be consistent with your perfect convenience, 

 as you then may be able to tell me in a moment the reason of my imperfect 

 success. 



At the close of letter x, Hamilton, writing on December i, fixed 

 December 4 as the day for exchanging confidences on the Wave Surface. 

 On that date accordingly Tait sent Hamilton his investigation along with the 

 following letter : 



Q. C. BELFAST, 



$th Dec. 1858. 

 My dear Sir William Hamilton 



I have to acknowledge the receipt of the rest of X with PS on two 

 separate occasions, also of pp. 17 28 of XIV....I shall take an early opportunity of 

 expressing my ideas with respect to V and X on the subject of finite differentials 1 

 meanwhile, as it is now late, I must explain as I best can the enclosed, which 



' This expression of ideas seems never to have been given. Other and more important 

 Quaternion developments had to be considered. 



