Bary centric Perspective. 1 77 



of the distance of the element from the plane of relation, and 

 OP, OR vary directly as the distance ; hence the frequency of 

 the atoms at any element in either sector will vary as the inverse 

 cube of its distance from the plane of relation, and hence this 

 will be the law of frequency for elements all over the plane, and 

 is irrespective of the particular positions of O, 0'; and conse- 

 quently, the density being proportional to the product of the 

 frequency of the atoms by their atomic weights, the law of den- 

 sity is that it varies about any point as the inverse fourth power 

 of its distance from the plane of relation. In like manner, by 

 taking three points 0, 0', 0" in the plane of relation and divi- 

 ding space into solid elements by plane bundles passing through 

 0', 0'', 0' 0" respectively, it may be proved that the law of 

 density for a solid figure will be that it varies as the inverse 

 fifth power of the distance from the plane of relation *. 



Atoms whose weights vary inversely as their distances from 

 the plane of relation may be termed like atoms ; lines, areas, and 

 solids whose elements vary in density inversely as the cubes, 

 fourth powers and fifth powers respectively, may be termed qua- 

 liform figures, or figures of qualiform density, the terms like and qua- 

 liform being adopted as the closest analogues to e^w«/and uniform. 

 It now becomes true, and may easily be verified, that the centres 

 of gravity of a qualiform finite line, triangle, and tetrahedron are 

 respectively identical with the centres of gravity of like atoms 

 placed at their apices f; and so every known or discoverable 



* The law of density for a solid is the inverse fifth power, for an area 

 the inverse fourth power, and for a line the inverse third power. Here we 

 must stop, for a point is that which has no parts : we can speak of the law 

 of atomic weights at a point, but not of density, for the latter implies the 

 existence of elements which are wanting to the point. In a hyper-ontolo- 

 gical sense there would be no objection to saying that for an element of a 

 point the law of density in this theory is as the inverse square, always 

 remembering that no such element exists. 



f As regards the finite line, these results may be very easily verified by 

 the integral calculus. For the triangle, it may be made to depend in the 

 preceding case by drawing from the point where the direction of any side 

 intersects the plane of relation, rays dividing the triangle into infinitesimal 

 portions ; the centre of gravity of every one such portion will easily be 

 seen to be in the right line joining the harmonic centre of the intersecting 

 side with the opposite angle; and an analogous method applies to the 

 tetrahedron. 



The same results may also be obtained analytically. Thus, ex. gr., for a 

 qualiform triangle whose apices are distant h, k, I from the opposite sides, 



and -» „» - from the plane of relation, the distances of the centre of gra- 

 vity from the respective sides will be 



hot fcft ly 



«+/3+y' «+j3 + y' «+j8+y' 

 The masses, say M, of a qualiform line, triangle, or tetrahedron, using a,, (B ; 

 Phil, Mag. S. 4. Vol. 26. No. 174. Sept. 1863. N 



