PROF. CAYLEY ON COMPOUND COMBINATIONS. 99 



be always at distances very nearly equal to the radius from 

 the vertex ; and, by what we have proved above, for some 

 position of the cone the attractions of the opposite sections 

 must be equal. Therefore (in fig. 2), for two distances 

 very nearly equal to VM the ordinates must be equal to 

 one another. Then the tangent to the curve near N must 

 be parallel to VX. But M is arbitrary; for we can take 

 the sphere of any size. Therefore at all points the tangent 

 to the curve is parallel to VX ; and therefore the curve must 

 be a straight line parallel to V X ; or, the attractions by 

 sections of the cone of equal thickness are constant, where - 

 ever the sections be taken. But the sections are propor- 

 tional to the direct square of the distance ; and therefore 

 the law of the attraction must be that of the inverse square 

 of the distance. 



XIV. On Compound Combinations. 

 By Prof. Cayley, F.R.S. &c. 



Read February 6th, 1877. 



Prof. Clifford's paper, ^' On the Types of Compound State- 

 ment involving Four Classes," relates mathematically to a 

 question of compound combinations ; and it is worth while 

 to consider its connexion with another question of com- 

 pound combinations, the application of which is a very 

 different one. 



Starting with four symbols, A, B, C, D, we have sixteen 



H 2 



