386 Mr. George J. Allman's Account of 



This can be easily proved by the help of the following geometrical 

 theorem : 



If from the extremities A, B, C of the three axes of an ellipsoid, three 

 parallel chords A/>, B^', Or, be drawn, and if these chords be projected each 

 on the axis from whose extremity it is drawn, then the sum of these three 

 projections, Aa, Bj3, C7, divided respectively by the lengths of the axes AA', 

 BB', CC, on which they are measured, will be equal to unity. 



Now conceive three chords Ap, Ap\ Ap", to be drawn from A, making 

 each with the other two very small angles, and so forming a pyramid with a 

 very small vertical solid angle u> ; and from B and C let two systems of chords 

 B^, Bq', Bq", and Cr, CV, Cr", be drawn, each system forming a very small 

 pjTamid whose three edges are parallel to the three edges Ap, Ap', Ap", of the 

 pyramid which has its vertex at A. 



The attractions of the three pyramids, reduced each to the direction of the 

 axis passing through its vertex, will be equal to fifpio.Aa, fxp/w.B^, fxfpw. Cy 

 respectively, and, therefore, the sum of those attractions divided respectively 

 by the lengths of the axes will be 



^ f Aa Bp Cy\ 

 fi^P''[AA'^BW+CC'r^^'""- 



CC'J 



Let pyramids thus related be indefinitely multiplied, and the ellipsoid will 

 be simultaneously exhausted from the three points A, B, C. 



Hence the sum of the whole attractions at A, B, C, divided respectively by 

 the lengths of the corresponding axes, will be ^Trufp, or, 



ABC,, 

 a b c ""^ 



Proposition IV. 



To find an eapression for the potential V of a system of particles at a point M 

 whose distance from the centre of gravity of the system is very great compared with 

 the mutual distances of the particles. 



It is proved by PoissON,* that if the origin of co-ordinates be at the centre 

 of gravity, 



* Mecanique, tomei. p. 178. 



