316 PROCEEDINGS OF SECTION A. 



Among other forms of F (r), for which the integral ^/J^' F (?•) dV 

 may be evaluated, are — 



F {}•) = ?•" log. r, n a positive integer, zero or — 1 

 F (r) = (log. r) - 



F (0 = ^^"^- ^"^ " 

 r 



§ 5. If in equation (2) we make F (r) =1, 



then ^1^ S ,IY = ^-^ jl 2 (U + .„y + „=) rfS. 



Making 2 := x™ and taking the integrals through the volume and 

 over the surface of the sphere previously employed, we obtain the 

 general integral — 



\\\ 



(m + 1) (m + 2) (ot + 3) ( ^ ^ 

 [(m + 2)a - b']-]- {b- a)'" * ' [(m + 2) « + &] j (19) 



This integral becomes indeterminate for the values m = — 1, or 

 — 2, or — 3. For these particular cases — 



III^-=-[--+--^:] (-) 



irj-L.--^-'-:-^:! ^-> 



§ 6. Let an anchor-ring be formed by the revolution about the axis of 

 z of the circle whose equation in the zj; plane is — 



(:C — b)'^ -j- 2^ = «-, 



where b y a. 



Let a plane be drawn through the axis of s, making an angle with 

 the plane xz. 



The co-ordinates of any point P on the surface of the anchor-ring 

 are given by — 



a; = ((5 + o cos. 6) cos. 



y =z {b -\- a cos. 6) sin. 



z = rt sin. 6, 



where 6 is tlie vectorial angle to P from the centre of the circular sec- 

 tion measured from the diameter of the section which passes through the 

 origin of co-ordinates. 



