RIGID BODY ABOUT A FIXED POINT. 287 



This may easily be put in the simpler form 



2d = Vdr, - V . (7 + £) V . (7 - jry* «*..'. (34). 



Reduced to scalars, this gives three linear differential equations of the first order, 

 the coefficients being functions of t. These can, of course, be reduced to depend 

 upon one linear differential equation of the third order with coefficients functions 

 oft. 



36. As a verification of the preceding work, we may try whether the result is 

 consistent, as it ought to be, with the condition (assumed throughout). 



Constant = (Tqf = 2y 2 b 2 + 2S . yblb . . 



This expression gives, by differentiation, 



=- PSyt, + 2( 7 2 - SyQSbb + ^SybSyb. 



Substituting for $ its value from (34), we have 



= - b^yt, + S . by^Sbr, + 2Syd(S . yh - is . (7 + £)*») 

 = — b^Sy^ + S . by'^Sbrj + SybS . ybr\ — S7SS . tbri 

 = _ S2g^ + g_ a{»jS.7^ + 5S.U73 + 7S.$i8} 



= - 5 2 S 7 £ + S.8(8S. 7 ^) 



which is true, because by (20) 



37. Another mode of attacking the problem, at first sight entirely different 

 from that in § 10, but in reality identical with it, is to seek the linear and vector 

 function which expresses the Homogeneous Strain which the body must undergo 

 to pass from its initial position to its position at time t. 



Let 



■sr — ya 



a being (as in § 19) the initial position of a vector of the body, ■& its position at 

 time t. In this case x is a linear and vector function. (Tait's Quaternions, 

 § 355.) 



Then, obviously, we have, ^ 1 being the vector of some other point, which had 

 initially the value a x , 



(a particular case of which is 



T-ar = T^a = Ta) 



and 



Vzrar 1 = V . X a X a l = % Vaa i ■ 



