134 PETER GUTHRIE TAIT 



In letter xvm, dated April 12, 1859, Hamilton returned to the 

 wave surface, and after deducing afresh its equation remarked : 



" Could anything be simpler or more satisfactory ? Do you not feel, as well as 

 think, that we are on a right track, and shall be thanked hereafter? Never mind 

 when ____ 



" De Morgan and I have long corresponded unofficially and said odd things to 

 each other. He was the very first person to notice the quaternions in print, namely, 

 in a paper on Triple Algebra in the Camb. Phil. Trans, of 1844. It was, I think, 

 about that time, or not long afterwards, that he wrote to me, nearly as follows : 

 'I suspect, Hamilton, that you have caught ttte right sow by the earl' Between us, 

 dear Mr Tait, I think that we shall begin the SHEARING of it." 



Tait replied in letter 31 of date April 13, 1859: 



I have just received XVII and XVIII, the latter an hour or two ago. 



Your deduction of Fresnel's construction from the symbolic form of the equation 

 to the wave is very elegant. I have given (in a paper which I suppose is now 

 being printed, for it has been sent off ten days or more) a proof of the same, which 

 is a mere interpretation of some of the equations which I have written down in 

 deducing that to the wave. 



I have recently (as I mentioned in letter 26) come to a seemingly formidable 

 difficulty in Quaternions. It is to find the most general form of linear and vector 

 function i/r from the equation 



where a- = (< a + p*)~V and where the scalar and vector constants of the required 

 function i/r involve p, a- and the operation <.... 



In the third PS. to your VIII you mentioned a result of Maccullagh's 1 which 

 I have since found in the Trans. R. I. A. I was lately trying the problem in an 

 extended form. I find for instance the following amongst a host of other results. 



(1) If the two lines which move in the planes are not at right angles, let the 

 cosine of their inclination be e, and let the third line be perpendicular to them ; it 

 traces a cone of the 4th order.... 



(2) If one of the moving lines be a generating line of a cone of the second 

 order, the second lying in a plane which passes through the vertex thereof, and the 

 third perpendicular to the other two, the locus is in general a cone of the 8th 

 order.... 



While this letter was being penned, Hamilton was beginning his 

 letter xix, the importance of which demands a full transcription. 



1 As given by Hamilton, the problem is, If three rectangular lines so issue from a common 

 origin that two of them move in fixed planes, the third will describe a cone of the 2nd 

 order, whose circular sections are parallel to the two planes. 



