CORRESPONDENCE WITH CAYLEY 157 



ix+jy + kz we obtain ix^+jy^ + kz^ the x^ y lt z^ being the new values of x, y, z 

 produced by the rotation: viz. the particular operation is 



ixi +jyi + kzi = q (ix +jy + kz) g~ l 



and you say that q( )f l is the rotation. But of course q, r being the two quater- 

 nions, qr in my mode of expression or qr( ) (qr)~ l in yours, belongs to the 

 resultant rotation. 



In my mode of expression 1 



^{cos i/+ sin kf(* cos a +/cos yS + k cos 7)} 

 is equally well with 



cos \f+ sin \f(i cos a +/cos /8 + k cos 7) 



the symbol of the rotation ; and my question was is there any interpretation, in 

 connection with rotations, of the sum 



T {cos i/+ sin \f(i cos a +j cos /8 + k cos 7)} 

 + T {cos \f + sin \f (i cos a' +/ cos ft 1 + k cos 7')}, 

 that is of the sum of any two quaternions 



w + ix +jy + kz, v/ + ix" +jy' + kz 1 . 



I think therefore you have understood me quite rightly viz. in asking about the 

 interpretation of a sum qua rotation, I do mean the effect of (q + r) ( ) (q + r)~\ and 

 as to the product qua force I do refer to Vqr and shall be much obliged for the 

 answer. 



Believe me, dear Tait, yours very sincerely 



A. CAYLEY. 



Nov. bth 



PS. I believe it was I who first gave in the Phil. Mag. the formula q(ix +jy + kz)q~ l , 

 showing it was identical with that of Rodrigues for the effect of a rotation but 

 Hamilton was doubtless acquainted with it. 



Tait replied to this the next day : 



7/1 1/82. 

 My dear Cayley 



The note I sent you yesterday, and which I hope you got, will now, 

 I see, more than answer your question ; which (as I understand it) refers to the sum 

 of two versors 



Uq+Ur 



1 There is a strong resemblance here between Cayley's symbolism of the rotation involved 

 in the quaternion and the discussion by Klein and Sommerfeld in their Ucber die Theorit des 

 Kreisels of what they call "die Quaternionentheorie " (Chap, i, 7). See Tail's paper "On 

 the claim recently made for Gauss to the Invention (not the Discovery) of Quaternions" 

 (Proc. R. S- E- Vol. xxm, 1889); and "Professor Klein's View of Quaternions, a Criticism," 

 by C. G. Knott (Proc. R. S. E. Vol. xxm, 1889). 



