1 72 Prof. Sylvester on the Centre of Gravity of 



It is clear that these results may be extended to space of the 

 higher dimensions. Thus in the corresponding figure in space 

 of four dimensions bounded by the hyperplanar quadrilaterals 

 abed, ufty8, which will admit of being divided into four hyper- 

 pyramids in twenty-four different ways, all corresponding to the 

 type 



a b c d a. 



b c d a ft 



c d a ft y 



d a ft y 8 



there will be a cross-centre given by the intersection of any four 

 out of twenty-four hyperplanes resoluble into six sets of four 

 each, — one such set of four being given in the scheme subjoined, 

 where in general pqr means the point which is the centre of 

 (p, g,r), and the collocation of four points means the hyper- 

 plane passing through them, viz. 



By8; y8a; Bab; abc ; 



y8a; Sab; ubc ; bca; 



8a ft ; aftc; ftcd; cdb ; 



afty; ftyd; yda; dac. 



The mid-centre will mean the centre of the eight angles a, b, c, d, 

 a, ft, y, 8, regarded as of equal weight ; and to find the centre of 

 the hyperpyramidal frustum, we may either produce the line 

 joining the cross-centre with the mid-centre through the latter and 



not as themselves, but as their own centres of gravity ! Some of my readers 

 may remember a signal case of a similar autometamorphism which occurred 

 to myself in an algebraical inquiry, in which I was enabled to construct 

 the canonical form of a six-degreed binary quantic from an analogy based 

 on the same for a four-degreed one, by considering the square of a cer- 

 tain function which occurs in the known form as consisting of two factors, 

 one the function itself, the other a function morphologically derived 

 from, but happening for that particular case to coincide with the func- 

 tion. This parallelism is rendered more striking from the fact of 4 and 6 

 being the numbers concerned in each system of analogies, those numbers 

 referring to degrees in the one theory and to angular points in the other. It 

 is far from improbable that they have their origin in some common principle, 

 and that so in like manner the parallelism will be found to extend in general 

 to any quantic of the degree 2n, and the corresponding barycentric theory 

 of the figure with 2n apices (n of them in one hyperplane and n in another), 

 which is the problem of a hyper-pyramid in space of n dimensions. The 

 probability of this being so is heightened by the fact of the barycentric 

 theory admitting, as is hereafter shown, of a descriptive generalization, de- 

 scriptive properties being (as is well known) in the closest connexion with 

 the theory of invariants. Much remains to be done in fixing the canonic forms 

 of the higher even-degreed quantics ; and this part of their theory may here- 

 after be found to draw important suggestions from the hyper-geometry above 

 referred to, if the supposed alliance have a foundation in fact. 



