442 Mr. J. J. Sylvester on the Rotation of a 



If in our figure the order of the rotations had been reversed, 

 PQr, QPr would have been taken respectively equal to PQR, 

 QPR, but in the opposite side of PQ, and r would have been 

 the resultant pole, the resultant rotation remaining in amount 

 the same as before. 



If either of the rotations had been negative, the resultant 

 pole would be found in QR produced, viz. at the intersection 

 ofrQ or HP with PQ. 



Calling the resultant rotation 7, we have always 



sin - : sin ^ : sin ~- : : sin QR : sin PR : sin PQ. 

 2 2 2 



When the component rotations are infinitesimal in amount, 

 R and r will come together in QP; the order of succession 

 of the rotations will be indifferent, and we shall have 



a : {3 : 7 : : sin - : sin - : sin - 



: : sin QR : sin RP : sin PQ, 



which gives the rule for the parallelogrammatic composition of 

 two simultaneously impressed rotations*. 



If, next, we consider the effect of rotations about two poles, 

 P and Q, fixed in space (supposing, as above, that they take 

 place first about P and then about Q), we must take PQr equal 

 to half the contrary of the rotation about P,and PQr to half the 

 direct rotation about Q (the angle being now taken positive 

 which was on the first supposition negative, and vice versa) ; 

 so that, retaining the original figure, the first rotation will 

 bring r to R, and the second carry R back to r ; showing that 

 r is the resultant pole, and thatt P'rP, the resultant rotation, 

 will be double the acute angle between Qr, rP, as in the former 

 case. 



To popular apprehension the important doctrine of uniaxial 

 rotation may be made intelligible by the following mode of 

 statement. Take a pocket-globe, open the case and roll about 

 the sphere within it in any manner whatever; then closing 

 the case, there will unavoidably remain two points on the ter- 

 restrial surface touching the same two points on the celestial 

 surface as they were in apposition with before the sphere was 

 so turned about in its case. 



It is right to bear in mind that the whole of this doctrine is 

 comprised in, and convertible with, the following easy geome- 

 trical proposition relative to arcs of great circles on any sphe- 

 rical surface, including the plane as an extreme case. 



* Compare Mr. Airy's Tracts, art. " On Precession and Nutation." 

 t P' is not expressed in the figure given. 



