156 PETER GUTHRIE TAIT 



UNIVERSITY OF EDINBURGH, 



6/11/82. 

 My dear Cayley 



Since I wrote you I have fancied that I ought to have sent you the 

 answers, even if I have misunderstood you. 



I. When we deal with a sum of two quaternions, from the rotational point of 

 view, the ratio of their tensor plays a prominent part. In fact 



<*+')( )(3 + rY* = (9r*Tr( )r->(qr-i)-* 

 where x is a scalar, which is to be found from an equation of the form 



a sin A 



- -i - = tan xA. 

 a cos A + I 



This seems an answer to your question " Is there any interpretation of the sum 

 qu rotation?" 



It is the rotation r( )r~ l followed by (qr~ f f( )(gr- 1 )-*. 



Of course it may also be put in the form (g~ l ry ( ) (g- 1 r)~ v followed by g( ) q~ l 

 where y is another scalar found from a transcendental equation. 



Compounding these it may also be expressed as 



which is more symmetrical. 



But it can also be expressed as 



q 1 r q 1 ( ) q~ l r^ q~ l . 



When / and m are found from two equations of the form 

 2 (a cos a + b cos 



= c sin 2/a sin m/3 + cos 2/a cos mft, 

 sin 2/a cos mft, 



l> sin ft 



2 a sin a 

 itsinft 

 all the quantities a, b, c, a, ft, being known scalars. 



Of course the number of such expressions is endless ; and I wait further light 

 from you. 



2. As to the product qua " force " (as you call it), we have 



V.qr^Sr. Vq + Vr.Sq+ V. VqVr 



so that the " force " of the product appears as the sum of three forces ; two of which 

 are multiples of the separate forces ; the other is a force perpendicular to both. 



In great haste, 



yours truly 

 P. G. TAIT. 



Cayley's letter of the same date which crossed this one was as follows : 



Dear Tait 



It is only a difference of expression : I say that 



q = cos ^/+ sin ^f{i cos a +_;' cos ft + k cos 7), 

 is the symbol of a rotation because operating in a particular manner with q upon 



