Fig. 6. 



278 Potential Problems connected with the Circular Arc. 



Proceeding to the limit of a plane lamina of breadth 26, 



T= ^-irpwPb 4 ', a known result. 



The calculation of the fluid pressures on the lamina is not 

 difficult, but the " end pressures " of the preceding paper 

 must be taken into account. The final resultant is a force 



27Tpa) 2 a i sin 4 - acting along the ?/-axis. The result is also 



deducible from the general equations of motion now to be 



briefly discussed. 



§ 5. Refer to moving axes O'X', O'Y', fixed with respect 

 to the lamina as in fig. 6. Let the 

 coordinates of 0' with respect to 

 axes fixed in space be x, y, and the 

 angle between O'X' and OX be 

 denoted by 0. A rotation 6 does 

 not disturb the liquid, and so con- 

 tributes nothing to the kinetic energy. 

 Denoting by U, V, the velocities 

 of translation along O'X', O'Y' 

 respectively, equation (18) of the 



preceding paper gives 



2T = 27rpa4\J 2 sm 2 ^(l'+ cos 2 |) + V*sin?gl. ill) 



(This has also been deduced from 2T=J<£cfyr.) The usual 

 methods now give the forces acting on the lamina. For 



brevity, denote by A, 2irpa 2 sin 4 5, and by B, 7775a 2 sin 2 a. 

 Then 



2T = (A + B)U 2 +AY 2 , .... (11') 



giving the forces and couple 



X=-(A + B)U + AV6M 

 Y=-AV-(A-rB)U0, > • • • (12) 

 L= BUV. ) 



These have also been deduced from the general pressure 

 equation. On forming the general equations of motion 

 from (11'), the consequences of the fact that T is independent 

 of are at once apparent, for these equations will be found 



