1879.] Mr. W. D. Niven. On certain Definite Integrals. 5 



flf 



dx dy dz _ M «> 3 \ df dg dh 



- rg -— — ^ ^ 1 ... (4) 



PQ o2i-(-3 2i + l!- </f + tfi + tf 



where M is the mass of the ellipsoid. Equation (4) gives the har- 

 monic expansion of a solid ellipsoid of unit density ; and it will be 

 observed that, in that form, it gives a very simple proof of Maclaurin's 

 theorem, since the operator 



a?— + Z> 2 — + c 2 — 

 df dg dh 



is unaffected by the addition of the terms 



W dg dh) 



h 



df dg 



By means of (4) it is easy to find what Thomson and Tait have 

 called the exhaustion of potential energy, due to the mutual action of 

 two solid ellipsoids, and then by spherieal harmonic analysis, to deter- 

 mine approximately the nature of the forces acting between them. 



Pursuing the same method, we next find that the potential at an 

 outside point (/, g, li) due to an ellipse of density 1, with its axes 

 2a, 2b, in the axes of co-ordinates x, y f is 



From (5) it is easy to pass to the potential of a magnetic pole 

 at /, g, h, due to unit current circulating in the boundary of the 

 ellipse, and thence to the expansion in harmonics for the mutual 

 potential energy of the currents in two elliptic circuits in any positions. 



Results similar to (4) and (5) can also be easily found for solid 

 parallel opipeds and for rectangular circuits. ■ 



The series here found cease to be convergent within certain regions 

 including the origin. The expansion of the potential in those regions 

 must be found by an independent investigation. 



It may be remarked that the harmonics in the form of differential 

 coefficients, given in (3), (4), and (5), may easily be expressed in 

 terms of the corresponding surface harmonics, according to the follow- 

 ing theorem : — 



Let r be the radius vector from the origin O to any point P, and let 



f(—, — , -—J be any homogeneous operator of the itih degree operating 



\dx dy dz/ 



upon the reciprocal of r, then 



^{sd% dy dz) r ^ ^ r i+1 ^(,dx dy dz)^ 1 ^' 



