58 PfiOCKEDINGS OF SECTION A. 



About 100 southern variables remain, which can be observed only 

 occasionally at the Peruvian Station of this observatory. Organised 

 and systematic observations of these stars by the astronomers of 

 Australia would supply a great want, and furnish a valuable contri- 

 bution to stellar astronomy. 



-NOTE ON A aEOMETRICAL ILLUSTRATION OF THE CONVER- 

 GENCE OF THE GEOMETRICAL SERIES. 



Bij PROFESSOR H. S. CARSLAW. Sc.D.. University of Sydney. jV.S.W. 



4.— ON CERTAIN SURFACE AND VOLUME INTEGRALS OF AN 



ELLIPSOID.— PAliT IT. 



By EVELYN G. HOGG. M.A., Chrigfg College. Christchurch, N.Z. 



The following paper is a continuation of one read by me before 

 the A.A.A.S. at its Melbourne meeting (1900).* This earlier paper 

 will be referred to as Part I., and for the sake of convenience the 

 integrals in the present paper will be numbered so as to follow on 

 consecutively with those given in Part I. 



I have proved elsewheref the following theorem connecting 

 surface and volume integrals of any closed surface, viz. : — 



C C C^ F {r) d r 



■=f.^^'^^^?^^"^[.^/^'^^^w<"■]'^«. w 



r 

 where g is a homogeneous function of degree '11: in x, ;/, and z\l, m^n 

 are the dirrction -cosines of the outward drawn normal to 8 at the 

 point oc,y, z: >" := cc" + y- 4- z'. The triple integral is taken through 

 the volume enclosed by 8 r.nd the double integ;ral over the surface 8, 

 and S and -F (/■) are subject to the condition that they do not become 

 infinite at any point on or within the surface 8. 

 Let ihe equation of the ellipsoid be — 



^ + y"" + ^J = 1, 



then Ix 4- vii/ + nz is the length of the central perpendicular on the 

 tangent plane to the ellipsoid at {x y z), and (?) may be written — 



f f If. [ f'- '' ^ ' i^ w ^r ]^ ^ - !\!\!\^ ^ ^''^ ^ ^ ^-^^ 



r 

 Lft F {r) := r '^" and g = / ^ + -^ ' + ~l \ ' 



V « ^ " '^' ' 



r 2/,- + 2 ' r "'■" "^ -" "" ' 



then 1 r F (r^ dr = -.-, ^ ?; and S is unit vat all pomts on 



./ ^ ^ 2k + 2n + S 



the surface uf the ellipsoid. 



* Proe. A.A.A.S. (Melbourne), IHOO, p. 191. 



t "On a form of (rreen's Tnedrein." Proc. A.A.A.S. (Adelaide), 1007, p. 312. 



