CORRESPONDENCE WITH CAYLEY 165 



In his reply Tait suggested that Cayley should communicate his note 

 to the Royal Society of Edinburgh, and then continued : 



Of course I do not agree with you, in fact we are as far as the poles asunder 

 in regard to your main point. There we must continue to differ 



I scarcely think you do me justice in giving without its context the remark [in 

 the Preface to Tait's Quaternions, ist, 2nd, and 3rd editions] " such elegant trifles as 

 Trilinear Coordinates." I think that you will see that the context very considerably 

 modifies the scope of the remark: so much so, in fact, that while I am still of the 

 opinion I expressed I am not prepared to use the phrase " elegant trifles " even about 

 Trilinear Coordinates (of Quadplanar Coords. I said nothing) without some such 

 qualification, or setting, as it has always had in my Preface. 



Cayley replied : 



CAMBRIDGE, lyh Junt. 

 Dear Tait 



Thanks for your letter. I am quite willing that the paper 'should be 

 read at the R.S.E. did you mean also that it should be published in the Proceedings? 

 if you did I am quite willing to let this be done instead of sending it to the Messenger. 

 Please make the reference to the preface of the ist as well as the 2nd and 3rd editions 

 and make any additions or explanations to show the context of the " elegant trifles." 

 I was bound to refer to quadriplanar coordinates, because the comparison is between 

 Quaternions, which refer to three-dimensional space, and the Cartesian coordinates 

 x, y, z or in place thereof the quadriplanar coordinates, x, y, x, w. Of course you see 

 my point. I regard the trilinear or quadriplanar coordinates as the appropriate forms 

 including as particular cases the rectangular coordinates x, y or x, y, z and bringing 

 the theory into connexion with that of homogeneous forms of quantics and as 

 remarked in my last letter, it is only in regard to these that the notion of an 

 invariant has its full significance ; so that trilinear coordinates very poor things, 

 Invariants a grand theory, is to me a contradiction. 



In a long formula of Gauss which you quote for its length, you give the expanded 

 form of a determinant the expression is the perfectly simple one 



x'-x, y'-y, z'-z -r (dist. P, Qf 

 dx dy dz 

 dx' dy 1 dz" 



Do you put any immediate interpretation on of = scalar, or consider it merely 

 as a necessary consequence of the premises ? 



Believe me, dear Tait, yours very sincerely 



A. CAYLEY. 



Cayley's paper On Coordinates versus Quaternions and Tait's reply 

 On the Intrinsic Nature of the Quaternion Method were published side 

 by side in the R. S. E. Proceedings, Vol. xx, pp. 271-284. As each of them 

 expressed it in the correspondence, they differed fundamentally. Cayley 



