460 



Lord Rayleigh on the 



two hemispheres, which the pressure is evoked to balance. 

 If the fluid on one side of the diametral plane extended to 

 infinity, the attraction upon the other hemisphere, supposed 

 to retain its radius r, would be irr^K. simply ; so that the 

 second term T . 27rr may be considered to represent the de- 

 ficiency of attraction due to the absence of the fluid external 

 to one hemisphere. Regarding the matter in two dimensions, 

 we recognize T as the attraction per unit of length perpen- 

 dicular to the plane of the paper of the fluid occupying (say) 



Fig. 4. 

 Y 



the first quadrant X Y (fig. 4) upon the fluid in the third 

 quadrant X' Y', the attraction being resolved in one or 

 other of the directions OX, Y. In its actual direction, 

 bisecting the angle X Y, the attraction will be of course 

 -/2.T. 



We will now suppose that the sphere is divided by a plane 

 A B (fig. 5), which is not diametral, but Fig. 5. 



such that the angle BAO = 0; AO = ^ 

 AB = 2p. In the inierior of the mass, 

 and generally along the section AB, 

 Y = 2K. On the surface of the sphere, 

 and therefore along the circumference 

 of A B, V = K - 2T/r. When V was in- 

 tegrated along the normal, from a plane 

 surface inwards, the deficiency was 

 found to be 2T. In the present application the integration 

 is along the oblique line A B, and the deficiency will be 

 2T sec 0. Hence when r and p increase without limit, we 

 may take as the whole pressure over the area A B 



7r/o 2 (K 4- 2T/r) -2wy> . 2T sec 6 



=7T P 2 K - 2tt/>(2T sec 0-T cos 6). 



The deficiency of attraction perpendicular to A B is thus for 

 each unit of perimeter 



2Tsec£-Tcos0, 



(41) 



