EULER'S EQUATIONS FOR A RIGID BODY 347 



Resolved along the axes OA, OB, OC, the first part has compo- 

 nents 



6 sin i/r, 6 cos ty, 0, 



while the second has components 



<j> sin 6 cos T/T, <j> sin 6 sin -v/r, < cos 0. 



Compounding these motions, we obtain 



tw 1 = 6 sin i/r < sin cos i|r 1 

 G> 2 = 6 cos A/T -f- $ sin d sin T/T L. 

 co 3 = ijr -f < cos 



Let the work done in a small displacement be 



80 + <&ty + ty; 

 then Lagrange's equation for the coordinate M* is 



on substituting from equations (185). Also 



dT da*. a 2 



- i 



-* 



We have, by differentiation of equation (184), 



j (0 cos T/T + <^> sin 6 sin i/r) 



+ ^o> 2 ( 6 sin ^r + sin cos 



Finally ^Sijr is the work done by external forces in a small 

 rotation &/r, and therefore, by 121, "9 is equal to N, the sum of 

 the moments of these forces about the axis OC. 



