124 PETER GUTHRIE TAIT 



unit vector. The question had reference to the finding the locus of ultimate inter- 

 sections of the series of paraboloids, a problem which arose out of an investigation 

 I was lately making and which I felt was too much for me at the time but if 

 you will permit me to withdraw it again for a little, I think I may perhaps manage 

 it now. ^ 



The problem will be found solved in Tail's Quaternions, 321 (3rd 

 edition), very much as Tait solved it in his letter 8. Hamilton was 

 greatly taken with the question and discussed the geometry of the envelope 

 at great length in his letters xi and xui. The envelope is a surface of 

 revolution of the fourth degree having the quaternion equation 1 



and this Hamilton proposed to call Tail's Surface. It is curious to note 

 that the first solution sent by Hamilton to Tait did not agree with Tail's. 

 By his first method of elimination, in fact, Hamilton introduced a " foreign 

 factor" in the form of a sphere. In the very short letter xn he writes : 



" Your investigation would look much better in print than my own ; for you see 

 that I take no pains, in this correspondence, to put any check on a natural tendency 

 to diffuseness & scarcely ever copy from a draught, although the style of the 

 composition would thereby be greatly improved, especially in the way of condensa- 

 tion. 



"It takes, you know, -more pains to write a short than a long letter, or essay, on 

 any subject : not that I pretend to have taken any pains with this short note ! but 

 I must tell you, some time or other, of its once costing me half a quire of paper 

 to write a note of one page to a lady who wanted my opinion on an astronomical 

 manuscript of her own." 



Meanwhile along with the prolonged discussion of Tail's Surface in 

 letter xui Hamilton was continuing his elucidation of the theory of 

 differentials in letter x. After acknowledging receipt of parts of these 

 letters on November 13, 1858, Tait continued in letter 10 in these 

 words : 



For a week I have been hard at work trying to deduce the equation to Fresnel's 

 wave-surface by a process purely quaternionic starting from the data employed by 

 Archibald Smith in the Cam. Math. Journal. As yet I have only deduced the 

 directions of the planes of polarization for any wave-front, and the law connecting 

 the velocities of the two rays, and these come out with admirable simplicity. In 

 attempting to find the equation to the surface I have come upon a terrible array 

 of Versors. Of the latter I have still a sort of horror arising principally I suppose 

 from my having avoided the use of them on any occasion on which it was possible. 



1 The Cartesian equation is <*"(*"+/) = (*'+/ + 



