Leathem — On Doublet Distributions in Potential Theory. 39 



with component moments A, B, C. A first approximation to the determina- 

 tion of the liquid motion would consist in an evaluation of A, B, C. We 

 shall show how the value of A depends on that of <f>. 



Apply Green's Theorem to the functions <b and x in the region bounded 

 internally by <S and externally by the sphere S' with centre at the origin and 

 radius R. Since in this region A<£ = 0, and A# = 0, the volume integrals 

 vanish, and we have 



t p~(RZ)dS' 



r dn 



RL%dS' + fx^dS. 

 dR J dn 



Now 



H^-*^-M*- M m 



dS' 



= - — L(AL + BM+ CN+ Q) d*>, 



4?rJ 



(where Q consists of negative powers of R multiplied by surface-harmonics 

 of integral orders greater than unity) 



= - A. 



Also, at S, dz/dn = I, when I, m, n are the cosines of the normal drawn 

 into the irrelevant region. Hence 



A = - \{xW - ty)dS. (13) 



14. Relation between douilet-effect at infinity and applied impulse, — The 

 formula (13) suggests consideration of the case in which the surface 8 is a 

 massless thin rigid shell having liquid inside as well as outside. Let <p' be 

 the velocity potential of the motion of the liquid inside the shell, and let dn 

 continue to represent the element of normal drawn away from the region of 

 (p and therefore into the region of </>'. Apply Green's Theorem to the functions 

 (f>' and x in the space enclosed by S ; we get 



\xdtfldndS = U'dx/dndS = \l<p'dS, 

 or \xWdS = \l$'dS. 



Substituting in (13) we get 



A = - j U$ - $) dS. (14) 



Now it is known that if the liquid motion were suddenly set up from 

 rest, as by the application of an impulsive forcive to the shell, the impulsive 

 liquid pressure set up would be - <f>. Consequently - jtydS is the a; component 

 of resultant impulsive pressure on the shell from without, and jl<b'dS is the 

 corresponding component of impulsive pressure from within. So the right- 

 hand side of (14) represents minus the x component of the resultant of the 



R.I.A. PKOC, VOL. XXXII, SECT. A. [6] 



