174 Prof. Sylvester on the Centre of Gravity of 



A.s we know that the centre of gravity in this case is in the 

 line joining the centres of the opposite faces, what is wanted 

 here is merely the proportion of the segments into which this 

 joining line is divided at the centre in question, or, in other 

 words, the ratio to each other of the distances of the centre 

 from the parallel faces. 



Let 



ab : a/3=bc : fty=ca :ya = l : X. 



Then obviously 



vol. abcct : vol. bca^ — aboi : bot/3 : / : X, 

 vol. bca.@ : vol. ca/3=ybcoc : cay : l:\: 



hence 



abcu : bcujS : cu/3y = l 2 : /X : X 2 ; 



also if h be the distance between abc, u{3y, the distances of the 



h h ^Kh 



centres of abcct, bca-ft, cotfty respectively from afowillbe -, -, — -. 



4 Z 4 



Hence the distance of the centre of the frustum from abc will 

 be ll / 2 + /X + X 2 >andsofroma/3 7 itwillbe^ /2 + a + ^ 2 J, 

 agreeing with the well-known formula applicable to this case*. 



But I pass on to a subject of much deeper interest. 



The geometrical constructions included in the preceding inquiry 

 (such for instance as depend on the properties of centres and oppo- 

 sites), like those which occur in the more ordinary theory of the 

 triangle and pyramid, at once suggest the existence of descriptive 

 propositions in which harmonic centres and harmonic opposites, 

 and in general harmonic multiplications and divisions, take the 

 place of the corresponding arithmetical operations. 



To make my meaning perfectly clear, let us conceive a fixed 

 plane ; and by a harmonic succession of points A, B, C, D . . . in a 

 line meeting the fixed plane f (which we may term the plane of 



* If we agree to denote by a,b,c; a, & y, the planes a/3y, byct, cuj3 ; 

 abc, fica, yah respectively, it may easily be shown that each quaternary 

 system of planes a, b,G6,fi; b, c, /3, y ; c, a, y, x passes through a single 

 point, we have thus given three points which determine a plane ; the inter- 

 section of this plane with the line a,b,c; a, /3, y is a sort of centre to the 

 frustum, and must possess properties deserving closer investigation. 



t It will of course be understood that in dealing with figures lying in 

 the same plane, a line of relation (viz. the intersection of the plane of rela- 

 tion with the plane of the figures) may be substituted instead of the former 

 plane, since the distances from the one and the other are in an invariable 

 ratio ; and so for different segments in a right line, we may substitute 

 a point of relation on the line itself instead of the plane. I deal with a 

 plane of relation as comprising implicitly all the subordinate cases; were it 

 required to go out into space of four or a higher number of dimensions, it 

 would of course become necessary to deal with hyper-planes of relation. 



