168 Prof. Sylvester on the Centre of Gravity of 



a b c a b c a ft c a b y 

 b c a ft c a ft y a b y a, 

 c a fty a fty ci b 7 oc ft 



b a c ft a c b a c b a 7 

 a c ft a. c b ay b ay ft 

 c ft a 7 b ay ft ay ft a 



If, then, taking any one of the above schemes we draw a plane 

 through the centres* of the three pyramids of which it is com- 

 posed, the six planes thus drawn will meet in a point, which will 

 be the centre of the frustumf. 



Let the point in which aa, ftb } yc meet when produced be the 

 origin of coordinates, and bcfty, caya, abaft be taken as the 

 planes of x, y, z ; and let 4«, 0, ; 0, 46, ; 0, 0, 4c be the co- 

 ordinates of a, b, c, and 4a, 0, ; 0, 4/3, ; 0, 0, 4y those of 

 a, ft, 7. Consider the first of the schemes above written. 



a h- a, by c will be the coordinates of the centre of abca, 

 a, b + ft,c „ „ „ bcaft, 



*, ft, C + y „ „ „ cafty; 



because, as everyone knows, the centre of a pyramid is the same 

 as that of its angles regarded as of equal weight. But again, 

 if we define as the mid- centre the centre of the six angles of the 

 frustum regarded as of equal weight, its coordinates will be 



2a + 2u 2b + 2 ft 2c + 2y 

 3 ' 3 ' c '' 



and if we substitute for each of the three centres last named 

 points lying respectively in a right line with them and the mid- 

 centre on the opposite side of the mid-centre and at distances 

 from it double those of these centres themselves, these quasi- 

 images of the centres in question will have for their coordinates 



, 2/3, 2 7 , 

 2a, , 27, 

 2a, 2b, . 



These points are accordingly the centres of the lines fty, ya, ab 

 respectively. 



And a similar conclusion will apply to each of the six schemes. 

 Hence using in general (p, g) to mean the middle of the line 



* I shall throughout in future for greater brevity hold myself at liberty 

 to use the word centre to mean centre of gravity. 



t I shall hereafter show that these six planes all touch the same cone, 

 of which, as also of its polar reciprocal, I have succeeded in obtaining the 

 equations. 



