with 



Brard 



AZg = 2 ^^B^^^ ^U 



E ^ok {fok(t') - f fok(^') <^kB(t'-r')dr'l 

 Lk=o I- ''o J J 



A51Ib= - 2^^bLU^ (^B^Lg-^) 



E ^ok fok(t') - J fok^^') 0uB(t'-r')dr' 



y (4) 



Obviously, the delayed circulation around the body gives -AZ*(0+) <0, 

 -AZjgCOi-) <0, -A51I*b(0+) <o, -A3n2B(0+) <0, and leads to an increase of the effi- 

 ciency of the plane. Because CLg is great with respect to a^ (see par. 5), the 

 effect of a diving plane located near the stern of the body may be very important 

 and shall not be neglected. 



7. Other Sets of Forces Exerted on the Body 

 (case of Par. 4) 



The constituents of the total set of forces were encountered at the beginning 

 of par. 5. In par. 5 and 6, we studied the forces due to the velocity potential of 

 the absolute motion. Let us now consider the other constituents. 



7.1. Forces Due to Gravity 



Let L^g.o.LCg be the coordinates of the center of gravity of the body, 

 L^^, 0, L^^ those of the center of the volume, ij. the density of the body with re- 

 spect to the fluid. 



We assume that the square of the angle of trim 6 is negligible. 



The components on the axis attached to the body of the forces and moment 

 due to gravity are 



(1) 



Xg = -pg W(/x- 1)5, Yg - , Zg=pgW(M-l), j 



7.2. Forces of Iner tia for the Body 



Let pWL^/xYj, pWLVx'2> /oWLV^'s be the moments of inertia of the body with 

 respect to axis acting through G and parallel to the axis 0(x,y, z). Let p^L^iJ.y:^^ 

 be the rectangular moment of inertia due to the product zx. 



864 



