176 Prof. Sylvester on the Principles of 



thing being true for equal atoms at distances dividing the line 

 into any even number of equal parts. Hence in the general ana- 

 logical theory we must take the infinitesimal intervals of our atoms 

 at points in harmonic succession. 



Let P, Q, R be any three such points, and let w, x-\-dx, 

 x-\-2dx-\-d 2 x be their respective distances from the plane of 

 relation ; and let q be the frequency at P, i. e. a quantity pro- 

 portional to the number of atoms which occur in a given infini- 

 tesimal space about P; then evidently qdx is constant, and 

 qd^x + dxdq = ; but by virtue of the harmonic relation between 

 P, Q, R, we have 



[x + 2dx + d q x) (dx) =x(dx + d 2 x) , 

 or 



xd*x=2(dx)*, ot-^=2— , 

 ' q x 



1 



l. e. q varies as -a* 



2 x l 



Moreover the weight of each atom varies as -, hence the density 



x 



of any element in a line must be taken to vary as the inverse 



cube of its distance from the plane of relation. 



Let us now endeavour to obtain the law of density for any 



element of a plane. Let 0, ; be any two points in the line in 



which the plane in question meets the plane of relation, and let 

 the plane be divided into infinitesimal elements similar to P Q S R 

 in the figure by pencils whose rays are in harmonic succession pro- 

 ceeding from and 0' ; then one atom belongs to every such 

 element, which will be the analogue of a rectangular element in 

 the common theory ; but the area of this element, as compared 

 with any similar element, say P'Q'S'R 1 in the infinite sector QOS, 

 varies as 



OP.RS + OR.PQ; 

 where P Q, R S, by what has been last shown, vary as the square 



