170 Prof. Sylvester on the Centre of Gravity of 



>ve written are the coordinates 



2<z + 2a 26 + 2/3 2c + 2 7 



which x, y, s above written are the coordinates. And the coor- 

 dinates of being 



3 



> 



3 : 



and those of G being 







A, 



B, 



c, 



it is obvious QOGr is a right line, and OG = JOQ, as was to be 

 shown. 



The analogy with the quadrilateral does not end here. There 

 is a construction* for the centre of a quadrilateral still easier 

 than that above cited, which may be expressed in general terms 

 by aid of a simple definition. Agree to understand by the 

 opposite to a point L on a limited line AB a point M, such that 

 L and M are at equal distances from the centre of AB but on 

 opposite sides of it; then we may affirm that the centre of 

 a quadrilateral is the centre of the triangle whose apices are 

 the intersection of its two diagonals (i. e. the cross-centre), and 

 the opposites of that intersection on those two diagonals respect- 

 ively. So now if we agree to understand by opposite points on 

 a limited triangle two points in a line with the centre of the tri- 

 angle and at equal distances from it on opposite sides, and bear 

 in mind that the cross-centre of a pyramidal frustum is the in- 

 tersection of either of two distinct ternary systems of triangles 

 which may be called the two systems of cross-triangles f, we may 

 affirm that the centre of a pyramidal frustum is the centre of a 

 pyramid whose apices are its cross-centre, and the opposites of 

 that centre on the three components of either of its systems of 

 cross-planes. This is easily seen ; for if we take the first of the 

 two systems, their respective centres will evidently be 



* This is the mode of statement (except that the important notion of 

 opposite points was not explicitly contained in it) which, accidentally meet- 

 ing my eye in a proof sheet of some Geometrical Notes (by an anonymous 

 author) intended for insertion in the forthcoming (if not forthcome) Number 

 of the Quarterly Journal of Mathematics, led to the long train of reflections 

 embodied in this paper, which but for that casual glance would never have 

 seen the light. The same construction, under another and somewhat less 

 eligible form, is given in the i Mathematician ' (a periodical now extinct, 

 edited by Dr. Rutherford and Mr. Fenwick, both of the Royal Military 

 Academy), 1847, vol. ii. p. 292, and is therein stated by the latter gentleman 

 to have, " as he believes, first appeared in the ' Mechanics' Magazine,' and 

 subsequently in the ' Lady's Diary ' for 1830." 



f From the description given previously, it will be seen that a cross- 

 triangle of the frustum is one which has its apices at the centres of either 

 diagonal of any quadrilateral face and of the two edges coterminous but 

 not in the same face with that diagonal. 



