280 



PROFESSOR TAIT ON THE ROTATION OF A 



= — jV^jj)?- 1 

 = - Y.qti<f>(n)q~} 



= - Y.q^- 1 q<p(q- 1 eq)q- 1 

 = - V. sp , 



which is the required expression. 



But perhaps the simplest mode of obtaining this equation is to start with 

 Hamilton's unintegrated equation (11), which for the case of no forces is simply 



But from 

 we deduce 



so that 



2 . mVzrzr = . 

 zs = \ sar 



sr= \ ezr + V £sr 



= are — aSfsr + v £33- , 



^ . m(V£srSszr — 1st 1 + zrSfarJ = 





2& 



— 1 



If we look at equation (16), and remember that <p differs from <p simply in having 

 ©• substituted for a, we see that this may be written 



Vspe + <pi = , 



the equation before obtained. The first mode of arriving at it has been given 

 because it leads to an interesting set of transformations, for which reason we 

 append other two. 

 By (17) 



therefore 



= qq-Kqfy- 1 + qtq- 1 - ffS? -1 ?? -1 , 



or 



= V 7 £ . 



But, by the beginning of this section, and by (14), this is again the equation 

 lately proved. 



Perhaps, however, the following is neater * 

 By (14) 



<pt = y . 



Hence 



<pi = — f>s = — & . m(-zrVz-sr + srVW 



= — 2/ . mis Sssr 



= — V. sZ . '/WzrSszr 

 = — Vepa . 



* [Inserted Dec. 19, 1868.] I have lately found that Hamilton, in his Elements of Quaternions 

 (1866), has obtained this equation in a manner almost identical with that last given. 



