a Truncated 



f Triangular Pyramid. 



4a 



"a* 



26 + 2/3 47 

 3 ' 3' 



4« 

 3' 



46 2c + 27 

 3' 3 ' 



2« + 2a 

 3 ' 



4/3 4c 

 3' 3* 



171 



Thus the three opposites to the cross-centre whose coordinates 

 are -2A + 2(a + «), -2B + 2(6 + /3), -2C + 2(c + 7), will have 

 for their x coordinates 



^-2« + 2A ; 

 o 



-2«+~J-2A, 



for their « coordinates 





"8 " 



-2/3 + 2B, 





-26 + 



¥ + ,B, 





26 

 3 



-f^B, 



and for their 



z coordinates 







2c 

 3 " 



-27+ 2C, 





-2c + 



| + *c, 





2c 



3 ~ 



-*+», 



and consequently the centre of the pyramid whose apices are the 

 cross-centre and its three opposites will be A, B, C, i. e. will be 

 the centre of gravity of the frustum, as was to be shown*. 



* I at one time supposed that a, b, c; a, ft, y formed two systems of 

 diagonal planes, and that there were thus two cross-centres; and dreamed 

 a dream of the construction for the centre of gravity of the pyramidal frus- 

 tum based upon this analogy, inserted (it is true as a conjecture only) in the 

 Quarterly Journal of Mathematics; but the nature of things is ever more 

 wonderful than the imagination of men's minds, and her secrets may be 

 won, but cannot be snatched from her. Who could have imagined a priori 

 that for the purposes of this theory a diagonal of a quadrilateral was to be 

 viewed as a line drawn through two opposite angles of the figure regarded, 



