PROCEEDIXGS OF SECTION A. 313 



If 6 be the angle between the normal to S at x, y, ::, and the radius 



\ector (r) to that point from the origin, then cos. 9 = - — — — ^— — — " ; 



/• 



also r/S COS. =: r" dw, where eho is the solid angle subtended at the 

 oritjin by the element of area ^S, hence the equations (2) and (3) trans- 

 form to — 



1,-^^ [i"""" F(r)rfrJ du, = j|J2F(r)f/V (2') 



l4? =0 (3') 



If in the above equations k is made zero, we obtain — 



\\ ^^±^^^J^ ■ / ,- F (.) dr] dS = Ijj F ir)dY (4) 



lx + my-\- nz ^g ^ Q ^^^ 



[ [/?- F (0 r/r] dw = 11 (" F (0 dV (4') 



{ dw = O (5' ) 



The limitations placed initially on the functions u, v, w require that 

 equations (3), (3'), (5). and (5') be restricted to cases in which the origin 

 of co-ordinates lies outside the surface S. In certain cases the equations 

 may be applied by imagining an infinitely small sphere drawn about the 

 origin of co-ordinates and taking the integrals over the original surface 

 and that of the sphere and through the volume bounded by those 

 surfaces. 



The subsequent part of the paper illustrates the application of the 

 above equations to the cases of the sphere and anchor-ring. 

 § 2. Let the equations of a sphere be — 



{x — b -) -\- ij- + z- = a-, 

 where b > n. 



TJie direction-cosines of the normal at the point x, y, z on the 

 sphere will be — 



X — b y z 

 a u a 

 and r" = a^ + 5- -I- 2ab cos.0, where is the vectorial angle to x, y, z 

 drawn from the centre of the sphere and measured from the diameter 

 passing through the origin of co-ordinates; also 

 (/S = 27r rt" sva.e de. 

 Hence equations (2) and (3) give — 



27ra r ■'"' ~ ^^ 2 r /V =^- ^ ^ F (r) dr ] sin.^ dd 



Jo r -*■ ' ^ "-• 



= ///'2F(.)^V (6) 



( ^' - \^ 2s[n.0de = O....{7) 



