THE LINEAR VECTOR FUNCTION 137 



but as I assumed that this condition was satisfied in most, if not in all, of the subsequent 

 theorems, I believe that I thought it convenient to enunciate it at starting. Besides 

 I wrote in some haste." 



Hamilton's letter xxm contains a systematic investigation of the linear 

 vector function, which differs markedly in the details of development from 

 the investigation given in his subsequent book The Elements of Quaternions. 

 In its initial stages it resembles Tail's mode of presentation, which Tait 

 himself calls "Hamilton's admirable investigation" (see Tait's Quaternions, 

 3rd edition, 156-159). Writing on May n, 1859, Tait in letter 33 

 remarked : 



Your No. XXIII (which I received yesterday) was indeed a treat. Nothing 

 could be more beautiful than your method of attacking the equation of the second 

 degree. I have been trying to supply for myself the demonstrations you suppressed 

 and have succeeded completely, though perhaps not elegantly. Thus as 



assume r" 1 \^ = m 



and if m = m', your theorem about the interchange of <f> and -fy is proved. The above 

 equations are evidently equivalent to 



<~ l Fi/r-'X/t = m 

 and m'^ V$\n 



Multiply together, and equate scalars, and we have at once 



m' (-S^V-S/tn/r-'X - \> 2 ) = m 

 or m' = m 



since Stj>\fj, = 



and therefore also S^~ 1 \;i = 



Another curious property of these functions resulting from this last equation is 

 that <^>~ 1 i/r is the conjugate of $-4f~*. 



I came upon the following (which seems neat). Generally, whether n be + or 

 or even = o, 



which is true (of course) of <f> also. 



What I was most puzzled with was the proof that m (in your notation) is a 

 constant. I saw at once that it could not contain the tensors of X and ft, but I did 

 not feel so sure about the versors. I have satisfied myself on that point by making 

 use of the distributive property of <f>~ 1 . 



Six days later in letter 34, Tait made a further reference to the same 

 investigation. 



T. 18 



