178 Prof. Sylvester on the Principles of 



theorem whatever relating to the centre of gravity of uniform 

 figures bounded by right lines or planes becomes immediately 

 transferable to that of qualiform figures of the same kind. Thus^ 

 to take a most simple example, since the centre of gravity of a 

 parallelogram is at the intersection of its diagonals, it must be 

 and is true that the centre of gravity of a quadrilateral whose 

 density at any point varies as the inverse fourth power at that 

 point from the line joining the intersections of its two pairs of 

 opposite sides, will also be at the intersection of the diagonals of 

 that figure. I am informed by Prof. Cayley that a somewhat 

 analogous consideration of altered density has been employed by 

 our eminent friend Professor William Thomson in his theory of 

 images, in reference to the distribution of electricity, given in 

 Liouville's Journal. 



It is an easy inference* from what has been established con-, 

 cerning the law of frequency , that if in the perspective of any 

 plane figure by tinting or relief we express the degree of crowd- 

 ed, j3, y; x,(3, y, § for the inverse distances of the apices from the plane of 

 relation, and V for the length, area, or volume, in the three cases respect- 1 

 ively become expressible under the very noticeable forms 



te±®«pV, *+P + V «$yt *+ P +y+ b *WY, 

 their moments in respect to the plane of relation being respectively 



M 



so that the mean density y is in each case a simple symmetric function of 



the atomic weights of the apices (it being of course understood that the 

 absolute atomic weight and frequency are each taken as unity). As the 

 same figure may be variously partitioned, and the sum of the component 

 areas and of their moments is unaffected by the mode of partition, the pre- 

 ceding formulae obviously give rise to, or imply the existence of, a class of 

 purely geometrical theorems relating to systems of points. It may be here 

 observed that the moment of a qualiform figure in respect to its plane of 

 relation represents the size, so to say, of (i. e. the number of atoms contained 

 in) the single molecule which, placed at the centre of gravity, will be the 

 statical equivalent of such figure ; for if n be this number, and d the distance 

 of the centre from the plane of the relation, and w the weight of the figure, 



. , . 1 , n 



since the atomic weight is -?, we must have -7 =w, or 



n—dw — moment of w in respect to the plane of relation. 



So in like manner, wherever the plane of relation is situated, two molecules 

 A and B, placed at two points, will be equivalent to the molecule A+B 

 placed at their centre of gravity. 



* It may here also incidentally be noticed that the area of the primitive 

 of any perspective projection of a figure in a given plane is proportional to 

 the attraction exercised upon it by the object plane indefinitely extended, 

 the force of attraction between any two elements being supposed to vary 

 inversely as the fifth power of the distance. 



