26 



^z(^') 



oo n-l 

 r. X F/zj- l< + rrp. ^ ^ Vrsj^a^^ < 

 n=l s=o 



, + 6^+1 ? - a^ a^+1 - h' ?^'\ ^^ ^ ^ + ^)' 

 n n n n ^ " ) {n - S - \)\ 



[60c] 



The total moment is then given by Equation [56]. 



Since the expression for ^Ji) is a bilinear form of the same type as that for ^ {i), 

 the discussion of the latter applies equally well to the moment. Hence, for continuous distri- 

 bution 



M 



i=jM/zJK,6^<,8;;rfT [561 



THE MOMENT DUE TO R0TATI0N,M3 



We define the moment \KJi) to be 



**3(0 = ( p/^w X r • n)(r X q) + (r X w • q)(r X n) + $[« X (r X n)i^da 



[61] 



and 



M, = I ^Ji) [62] 



'3 '-' "3^ 



The first two terms in the integrand can be reduced as a triple vector product, 

 (r X w . n) (r X q) - (r X w • q) (r X n) = r X [Cr X ui) X (q X n)] = (r X w ) (r ' q X n) [63] 

 since r • r x &> = 0. Making use of substitution [33], 



(r X «)(r • q X n) = (r. X w)(r. • q X n) + (R X ft»)(r. • q X n) [64] 



But by Equation [43], 



I (r. X w)(r. • q X n)c?a = (r. X u»)r- • j qxn(fCT = [65] 



Also, since it is evident that (R x ta){ii • <\ >^ n)dj involves only A, we have, using 

 Equation [35] 



