312 PROCEEDINGS OF SECTION A. 



3.— ON A FORM OF GREEN'S THEOREM. 



By HFELYX G. HOGG, M.A., Christ's College, Christchurch, Neio Zealand. 



§ 1. If S be a closed surface, I, m, and n the direction-cosines of 

 the normals to S at the point [x y z), and if u, v, and tv be, with their 

 first derivatives, finite, continuous, single-valued functions of the variables 

 X, y, z, then by a modification of Green's Theorem 



||(., + », + „.).S = |jj(l>| + '^).V, (1) 



where the double integ-ral is taken over the surface S, and the triple 

 integral through the volume enclosed by S. 

 Now let u — X ^f{r) 

 v = y^f^r) 

 iv=.z^f{r) 

 where 2 is a homogeneous function of degree 2k in the variables x, y, z, 

 and r^ z=z x" ■\- y" -\- z" . 



Then J = 2/ 0-^ + ^ 5 / (0 + f 2 1^, 



M^ith simuar expressions lor — and 



dy dz 



Hence f,^ + - + t= Sj (2.+ 3) /(,■) + >■ :J{ I 

 Hence (1) takes the form — 



^\^ {lx^my + nz)^f{rJdS = 



III 2 I {2k + 3) / (r) + .•!■{ .^V. 



Now let r'^ ^ {2k + 3) f {r) = F (r), then 

 dr 



f {?•) = Cr - (2* - 3) _|_ ^ - (2i . : ) f ^ 2W 3 p (-^-) ^^ 



where C is a constant, and 



p {{ 2 {Ix + mv + nz) f^S ff 2 {Ix + my + nz) 

 ^ JJ r-^*"^"^ "^ J) ^^"^^ 



C 



{ r-'^-F (r) 



fl'r 



rfS = 2 F (r) (/v. 



Now since the iritegral multiplied by C is independent of F (r), it 

 must vanish identically — hence we obtain the following equations: — 



n :^{lx^my + nz) H , ^-'^ . ^^ Y {r) drl dS 



l\ 



J 

 = j|J2F(r)./V (2). 



2 {Ix + my + nz) , 



— ^ rWTS f^S = O (3;, 



