Prof. Donkin on the Geometrical Theory of Rotation. 191 

 which, compared with the expression on the left side, gives 

 n/d\ dp, dv\ 

 "P^A-di^'li-'^di)' 



with symmetrical expressions for q and r. 



For a deduction of these expressions by direct differentiation, 

 see Mr. Cayley's paper in the Cambridge Mathematical Jom-nal, 

 vol. iii. p. 237. It is to be observed that the method employed 

 above does not require a knowledge of the expressions for X, //., v, 

 or p, q, r, in terms of the nine direction cosines of one set of 

 axes refen'cd to the other. 



Lastly, let it be required to find the relations between \, fi, v, 

 and the ■&, (j), ^jr of the ordinary theory. For convenience of de- 

 scription, let the earth be the body considered, the axes of x and 

 y being fixed in the plane of the equator, and that of z coinciding 

 with the polar axis, whilst the axes of ^ and t] are fixed in the 

 plane of the ecliptic. Then d is the inclination of the equator 

 to the ecliptic ; <^ is the right ascension of the axis of x ; and yjr is 

 the longitude of the ascending node of the equator, reckoned 

 from the axis of ^. 



Now it is obvious that the earth might have been brought from 

 the position in which the two sets of axes coincide, into that in 

 which it actually is at any instant, by three successive rotations ; 

 namely, fii'st, a rotation ^ round the axis of z ; second, a rota- 

 tion d round the axis of x ; and thii-d, a rotation ^ round the 

 axis of z again. Hence the principles above established give the 

 following equation : 



/c~4(l — fX— J/i — Av) = (cos^ — Asin^ ] (cos^ — f sin^ j 



(^cos|-/tsin|), 



of which the second side developed and compared with the first, 

 gives 



« = 6ec^ q • sec^ n i 



1^ — <f> li" — <f> 



(1 cos "^ „ ^ 51 sm ^ ,, ^ 



\=tan7r* ix=i&u-rr- , 



2 -ylr+d)' 2 'slr + A' 



cos ^ ^ ^ cos ~~ 



2 



V^ + 

 v= tan t, • 



The inverse expressions are easily obtained ; thus we have 



