a Truncated Triangular Pyramid. 



173 



measure off three-fifths of the distance of the joining line on the 

 part produced (as in the preceding cases we measured off two- 

 fourths and one-third of the analogous distance), or we may take 

 the four opposites of the cross-centre on the four components of 

 any one of the six systems of hyperplanar tetrahedrons of which 

 it is the intersection, and find the centre of the hyperpyramid so 

 formed. The point determined by either construction will be the 

 centre of gravity of the hyperpyramidal frustum in question. And 

 so on for space of any number of dimensions. It will of course 

 be seen that a general theorem of determinants* is contained in the 

 assertion that for space of n dimensions there will be 7r(n) quasi- 

 planes all intersecting in the same point, as also in the general 

 relation connecting this point (the cross-centre) with the mid- 

 centre and centre of gravity, of each of which it is easy to assign 

 the value of the coordinates in the general case. 



But returning to the case of the ordinary pyramidal frustum, 

 the preceding results lead at once to an easy geometrical proof 

 of the well-known analytical formula for finding the centre of 

 gravity of a pyramidal frustum in the case where the base and its 

 opposite plane are parallel. 



* We learn indirectly from this how to represent under the form of de- 

 terminants of the zth order, and that in a certain number of different ways, 

 the general expressions 



and 



(Z 1 Z 2 ...Z.-X 1 X 2 ...X.)< 



?iX 1 (Z 2 ?3. 1 .^-A 2 X 3 ...X.XZ 1 Z 2 ...?.^A 1 X 2 ...X.)»-2, 



a strange conclusion to be able to draw incidentally from a hyper-theory of 

 centre of gravity ! Thus, ex. gr., on taking i=4, we shall find 



bed cda, daft afty 



-.{abed- aftyhf 



:aa(bcd-ftyb)(abcd- aftyh)*. 



ftyd cda daft afty 



byb yha dab aby 



bed cda daft abc 

 And again, 



ad(bc + cft-\-fty) edec daft afty 



fta(cd-\- dy+yb) cda daft afty 



yb(da + a$ + 8ot) yda dab aby 



bc(ab-\-ba-\-aft) cda, haft abc 



The number of these representations will not be twenty-four, i. e. n(4), 

 but only twelve, the half of that number, because it will easily be seen that 

 the cycles abed, aftyd will lead to the same determinants, only differently 

 arranged, as the cycles beda, ftyba. I believe the law is, that the number 

 of varieties of such representations is ir(i), or -i-7r(i), according as i is odd 

 or even. The expression ab — aft at once conjures up the idea of a deter- 

 minant. We now see that there is an equally natural determinantive 

 representation, or svstem of representations, of (abc — aftyf, (abed— af 

 &c. 



