40 Proceedings of the Royal Irish Academy. 



impulsive pressure on 8 from both sides. Hence if X be the component of 



the impulse applied from without to the shell S which would produce the 



motion, our result is simply 



A = X. (15) 



The simplicity of this result, independent as it is of the size, form, and 

 position of S, and of all impulsive couples, is remarkable. 



An alternative statement of the same result may be got by considering, 

 instead of a rigid shell, a rigid body of the same density as the liquid, bounded 

 by the surface S. For this case we put 



W = l(u - y<j) Z + so) 2 ) + m (v - «w, + «w 3 ) + n (w -xwz + yu\), 



where («, v, w, w u w 2 , u> 3 ) is the velocity system of the rigid body referred to 

 the origin as base-point. It is then clear that 



* WdS = - f J (w - yu) 2 + 2o> 2 ) dx dy dz 



taken through the volume of the solid. Thus, in fact, \xWdS is minus the 



x component of momentum of the solid of unit density. The component X 



of the impulse necessary to set up the motion of solid and liquid has to 



supply the x momentum of the solid and to counterbalance the x component 



of pressure of the liquid, and so in this case also the equality (13) is 



equivalent to 



A = X. 



15. Approximate form, at great distance, of the potential due to a double- 

 sheet. — Eeturning from the bydrodynamical illustration to pure potential 

 theory, let us inquire into the approximate form at great distance of the 

 potential due to a double-sheet of variable strength r. A formula giving the 

 approximate form of the potential due to a system of positive gravitating 

 matter is well known, and it is natural to look for a corresponding expression 

 for doublet distributions. 



If we take any origin and let (§, v, Z) be the coordinates of a point of the 

 double-sheet and (I, m, ?i) the cosines of the normal, the potential V at a 

 point distant B from the origin in the direction (L, M, iV) is 



V = Jrr- 3 S/(Zi2 - %)dS, (16) 



where r is the distance from dS to the point (LB,, MR, MB). 



If B is very great we may get an approximate value of r' 3 by a binomial 



expansion thus — 



r 2 = B? - IB S (LI) + S£ 2 , 

 and therefore 



r" 3 = BT*\l - 2R-> S(Ll) + IV- 2£ 2 }- 3 '*, 



= B-*[l + 3B- 1 2(Z£) + smaller terms]. 



