23 



which is analogous to Equation [15]. The volume integral can again be transformed into a 

 surface integral, 



/p(rxq)c^T=p yx(r$)c?r=p (p{rxn)da 



[52] 

 s+s' 



With this step the correspondence between the two developments stops, for the surface inte- 

 gral in Equation [52] cannot be transformed by means of Green's reciprocal theorem, as was 

 the surface integral in Equation [16| . 



Collecting the results in Equations [491, [50], ^1], and [52], we have 



^S=\ P y(q • iXr X ") - (q • n)(f ^ q) da +j- \ p<^{x y<n)da 



+ 1 p<(w X r • n)(r X q) + (r X w ' q)(r X n) + $ [w X (r X n)]>fi(CT 



[53] 



This is again divided into three components; the first would be the moment if the flow were 

 steady (Lagally moment), the second arises when the flow is changing with time, and the last 

 is an additional effect due to rotation of the body: 



Mj =J p|l(q . q)(r xn) -(q • n)(r xq)|ja [54a] 



M^ =-^ r p1>(r X n)^^ [54b] 



M3 = p <!(w X r • n) (r X q) + (r X w • q) (r X n) + (t>[w x (r x n)'\^da [54c] 



While the component M cannot be reduced, it is a linear function of $, allowing the 

 superposition of solutions. It should be noted that the components M , M„, M, do not corres- 

 pond exactly to the forces F,, F. , F, since the integral for the force corresponding to M„ was 

 broken up into two parts, one becoming part of F, and the other part of F,. 



THE "LAGALLY MOMENT", M^ 



We again suppose the singularities to be discrete and isolated. The moment ^.(i) is 



then 



^ [55] 



^i(i)=\ pfeq • q)(r X n) - (q . n)(r x q) 



