Leathkm — On Doublet Distributions in Potential Theory. 41 



Consequently, if we neglect terms of a higher order of smallness than B' 3 , 

 V = fi- 2 \r(Zl + Mm + Nri)dS 



+ R-' j r j 3 (LI + Mm + Nn) (LI + Mr, + N£) - (H + m„ + nZ,)} dS, (16) 



in which formula we notice in passing the parallelism between the second 

 term and the expression for the mutual energy of two magnets. 

 Formula (16) is equivalent to 



V =• R- (pL + qM+ rN) + R~ 3 {(a - hb - §c) L- + (b - \c - U) W 



+ (c - \a - \l) N= + 3/MJSf + 3gNL + 3hLM\ (17) 



where (p, q, r) = [r(l, m, n) dS (18) 



and 



(a, b, c,f, g, h) s (t {211 2m,, 2< (ml + nr,), (nl + It,), (h + m£)) dS). (19) 



It is clear that, of these constants for the double-sheet, (p, q, r) are 

 independent of the choice of origin. Let us call (a, b, c) the moments of 

 inertia and (/, g, h) the products of inertia of the double-sheet with respect 

 to the coordinate planes, and let us examine how the values of the moments 

 and products depend on the positions of these planes. 



First let us change the origin, putting £ + £ for 5, &e., and accenting the 

 corresponding symbols for moments and products. Then 



a' = 2 j rl (£ + £„) dS = 2% p + a, (20) 



/' = jr{m(£ + £«) + n(n + ,„)} dS = Z, a q + n r +/. (21) 



These equalities are analogous to the theorems of parallel axes for ordinary 

 moments and products of inertia, but they do not indicate that any particular 

 origin has minimum properties analogous to those of the centre of gravity. 



If instead of changing the origin we take orthogonal axes of coordinates 

 in new directions, namely those whose cosines are respectively (A,, /u u vi), 

 (X.2, fit, v 2 ), (A 3 , ju 3 , v 3 ), we must write ZAi + mu. x + nv-^ for /, £A, + );/.(, + £v, 

 for £, and similarly for other cosines and coordinates. We thus get 



«' = «Ai 2 + bfi! 2 + cvr + 2ffi l v l + 2gv x \i + 2/iAi/ii, (20) 



/' = aA 2 A 3 + bfi.uz + ciw + (M2V3 4 v 2 fji 3 )f + (v 2 A 3 + A 2 v 3 )# 



+ (A 2/ u 3 + ju 2 A 3 ) h. (21) 



These relations are of the same form as those which hold good for 

 ordinary moments and products of inertia. Consequently it is possible to 

 choose, for any origin, such a set of orthogonal coordinate planes (principal 

 planes) as shall make the products vanish. For such planes the nine terms 

 of formula (17) reduce to six. 



It is of course clear that the discussion of this Article applies to any 



[6*] 



