158 PETER GUTHRIE TAIT 



(You write a T instead of a U\ but the form you adopt viz. 



cos a + (il+jm + kri) sin a 



is a versor, its tensor being i). 



Of course in this particular case, the formula I gave you yesterday is much 

 simplified. For instance we have a = i and x=\. 



Thus (Vq+Vr)( ) ( Vq + Vr)~ l = (qr~^r ( ) r~* (qr*$. 



This and indeed the general cases of q + r, is easily seen by means of a diagram 

 [proof given by use of a spherical triangle].... 



I send with this a copy of an old paper of mine bearing on the question raised 

 in your last... The second of these gives the reference which shows that Hamilton 

 anticipated you about the quaternion rotation. 



The third passage refers to what I thought was mine (i.e. putting Rodrigues' 

 expressions in a simpler form) but your letter shows that you also use this versor 

 form.... 



Cayley's reply was : 



Dear Tait 



Best thanks for the last two letters and the memoir. I am rather glad 

 to find that the formula was first given by Hamilton. 



The (g + r)( )(q + r)~ l formulae are very curious, but I hardly see as yet 

 what to make of them.... 



CAMBRIDGE, 8/A Nov. 1882. 



Cayley seems to have forgotten to some extent the contents of Tail's 

 paper of 1868. 



Towards the end of 1884 an interesting correspondence arose between 

 Tait on the one hand and Cayley and Sylvester on the other in regard to 

 the solution of the quaternion equation aq = qb. Sylvester had just published 

 his general solution of the linear matrix equation ; and taking a more general 

 view of the quaternion q he obtained what seemed at a first glance to be a 

 different solution from that given by Tait in his Quaternions. The analytical 

 theory which admits the possibility of Tq vanishing a possibility never 

 considered by Tait is given by Cayley in Chapter vi of the 3rd edition 

 of Tail's Quaternions ; and parts of this contributed chapter are almost 

 identical word for word with portions of Cayley's letters. 



On August 28, 1888, Tait in view of the preparation of this 3rd edition, 

 .asked Cayley for suggestions in the way of improvements, especially on the 

 analytical side. Cayley responded immediately with some notes which Tait 

 gratefully accepted. Some weeks later Tait wrote: 



