322.] BODY WITH FIXED POINT. ^5 



As, according to the fundamental property of the momental 

 ellipsoid (see (16), Art. 266), we have q e 2 /p\ this becomes 



/ .' (9) 



On the other hand, by Art. 258, if , & 7 be the direction- 

 cosines of /, z>. of the rotor a), we have 



hence 



T=$(A<* + Bto*+ Ceo*- 2 Da> y co z -2 Eco z co x -2 Fco x co y ). (10) 



Differentiating with respect to co x , co y , co z , and comparing with 

 the equations (2), we find 



-= A m -F,- .=# 



da) x 



H,, (ii) 



y 



If these relations be multiplied by a) x , co y , w z and added, they 

 give 



2 Tco^co.+co^H^+tf^ + ff^. (12) 



Substituting in the last expression aco = co x , /3co = co y , yco = co z , or 

 \H=H X , pH=H y , vH=H z , where X, jj,, v are the direction- 

 cosines of the vector H, we find 



We have, therefore, for the projection of H on the instan- 

 taneous axis /, 



2 T 

 H cos (f> = f? x ct. H- ff y j3 + H.fl = = Mcp'to j 



CO 



for the projection of o> on the direction of the vector H, 



2T 9 o) 2 



z - 



H H 



and finally, T= J //o> cos </>. (14) 



