THE LINEAR VECTOR FUNCTION 135 



SV, 

 April itfk, 1859. 



My dear Mr Tait 



Although what I am about to write must be very short, and might be 

 marked as PS. to No. XVII, yet, on the whole, I choose to number it as above, 

 partly with a view to encourage myself to write short letters. 



[i.] There is, as you know, a very important problem of transformation, to 

 which you have alluded, both in early and in recent letters, and of which I by no 

 means deny that those letters may contain a sufficient solution or solutions : for 

 I have hitherto avoided to examine them, in connexion with that problem, which 

 I certainly conceived myself to have resolved, about ten years ago, and to which 

 (as solved) I alluded at the end of art. 567, in page 569 of the Lectures.... 



[4.] The problem... haunted me, as it happened, yesterday, while I was walking 

 from the Provost's house to that of the Academy, &c. ; and _though I wrote nothing 

 down that day I resumed it this morning : and arrived at what you might call, in the 

 language of your No. 19, a ' perplexingly easy' solution (in the sense of being very 

 UNLABORIOUS, for I do not pretend that the reasoning does not require a close attention) ; 

 not in any way introducing ij k, nor a /3 7 (of an ellipsoid) nor t, K, but depending 

 entirely on the properties of the function (f>. So simple does this solution appear, 

 that I hesitate as yet to place entire confidence in it ; and therefore, till I have fully 

 written it out for at present it is partly mental and have given it a complete and 

 thorough re-examination, I hesitate to communicate it to you. Meantime, however, 

 I must say, that I am not conscious of having taken any hint, in this investigation, 

 from any of your letters.... 



[5.] April isth I shall just jot down here the enunciation of a few Theorems 1 , 

 which I have lately proved (as I think) anew, and which are intimately connected 

 with the question. 



THEOREM I. If $p be a distributive and vector and real function of a real 

 vector p, such that Sa<pp = Sptjxr, (a), then the eq n Vp$p = o, (/9), is satisfied by 

 (at least) one real direction of p. 



THEOREM n. Whatever be the given and real dir ns of p, (at least) two real and 

 rectangular directions, p and p", can be assigned, for a vector ra-, which shall satisfy 

 the two eq ns Spvr = o, (7), and Sp-nfasi o, (S). 



THEOREM III. If p and TO- satisfy the system of the three eq ns , (/3) (7) (&), 

 then w satisfies (/9), or more fully FUJ-^CT = o, (e). 



THEOREM IV. (Extension of Theorem I.) The equation (/3) is always satisfied 

 by at least one system of three real and rectangular directions, p lt p.,, p,, of p. 



Proof obvious, from what precedes. 



THEOREM V. The functional symbol <f> satisfies a cubic equation, 



whereof the three roots are always real. 



1 This is probably what Tait referred to in his paper on the intrinsic nature of the 

 quaternion method (1844; Sd. Pap. Vol. n, p. 396), where he states that "one of his 

 many letters to me gave, in a few dazzling lines, the whole substance of what afterwards 

 became a Chapter in the Elements" 



