272 



PROFESSOR TAIT ON THE ROTATION OF A 



12. The essential want of symmetry, in the system of three angles usually 

 employed, has led me to try various other systems. None of them, however, 

 were quite symmetrical, and I therefore introduce only one of them here. 



Suppose the position of the body to be determined by the angles \j/, 0, <p, 

 through which it has been made to turn about three rectangular axes which are 



fixed in it ; and which may be considered as - / ^dt, - I & 2 dt, j I u..dt respec- 

 tively; c^, « a , o> 3 having values in general different from w v « 2 , w 3 , but easily 

 deducible from them. 



The essential difference between this process and the ordinary one (just 

 treated), consists in using rotations about each of the three axes fixed in the 

 body, instead of one about one axis, followed by another about a second, and 

 then a final rotation about the first axis instead of the third. 



We have first a rotation ^ about i, next 9 about the new position of j, and 

 finally <p about the final position of k. 



fa ( ) i ~ •*- is the operator due to the rotation about i. It converts j into 



j) = j cos 4* + k sin -4/ 

 and k into 



h cos -v}/ — j sin 4* . 



Next, the operator due to the rotation 6 is 



and this converts k cos ^ — j sin \^ into 



£ = i sin 6 + (k cos *\> — j sin -vj/) cos 6 . 



Thus 



<P e ^ 



gsj'f i v = (cos | + 5 sin | J (cos - + » sin -J (cos J + i sin "| V 



