CORRESPONDENCE WITH CAYLEY 163 



" But then what do you mean by the vector a ? I mean 



so that the knowledge of a implies that of the coordinates.... 



" But your claim for the superiority of quaternions rests, as I understand it, on 

 the non-necessity of any explicit use of the equation in question a=*ix + jy + kz, or 

 a = iSia jSja kSka.. . . 



" As to the modus operand!, if in regard to the points (x, y, z) and (x 1 , y 1 , z 1 ) one 

 has to consider the combinations yz'y'z, zx' z'x, xy' x'y, I consider these directly 

 as the minors of the matrix 



x, y, z 



x, y', z" 



whereas you represent them (in what seems to me an artificial manner) as the 

 components of Fa/3.... " 



Tail's reply I give in full, since it presents in the briefest possible form 

 the fundamental principles of quaternions as Tait regarded them. 



38 GEORGE SQUARE, EDINBURGH, 



19/6/94. 

 My dear Cayley 



In the very first paper I published on Qns. (Mess. Math. 1862) I said 

 "the method is independent of axes... and takes its reference lines solely from the 

 problem it is applied to." Unless under compulsion, I keep to a, and do not write 

 either ix+jy + kz or iSia &c. 



Hamilton said (Lectures, p. 522) " I regard it as an inelegance, or imperfection, 

 in quaternions, or rather in the state to which it has hitherto been unfolded, whenever 

 it becomes or seems to become necessary to have recourse to " x, y, z, &c. 



Unfortunately like all who have been brought up on Cartesian food I now 

 and then think of a as iSia &c. (Hamilton himself was a terrible offender in 

 this way: his i, j, k, was almost a fatal blot on his system). But I know that 

 I ought not to do so, because a better way is before me. Thus : 



(P.S. What follows is, I see, Prosy. But it is necessary.) 



Position is essentially relative (though in physics direction may be regarded as, 

 in a sense, absolute) so we must have an origin, p then, or OP, I look on as P O, 

 the displacer which takes a point from O to P. Should it subsequently be displaced 

 to Q we have 



Hence all the COMPOSITION laws of Vectors. And of course the notion 

 of repetition of any one displacer, so that we get the idea of the tensor, and of the 

 unit vector. 



To COMPARE vectors, we may seek their quotient, or the factor which will 

 change one into the other. There are two obvious ways of looking at this. 



(a) The first is mathematical rather than physical. Here we introduce the 

 idea of a factor, such that 



a/p X p = <r. 



21 - 2 



