On the Centre of Gravity of a Truncated Triangular Pyramid. 167 



even when it is required to estimate the differences of conducting- 

 power which are brought about by alterations of temperature. 



2. That the conducting-power for good conductors, such as 

 copper, undergoes variations which are capable of comparison 

 with those which worse conductors, such as iron, present ; and 

 therefore 



3. That the proportionality which has been assumed to exist 

 between the conducting-powers of bodies for heat and for elec- 

 tricity is probably a fact. 



XXIV. On the Centre of Gravity of a Truncated Triangular 

 Pyramid, and on the Principles of Bary centric Perspective. By 

 J. J. Sylvester, F.R.S., Professor of Mathematics at the 

 Roy al Military Academy*. 



f~l ^HERE is a well-known geometrical construction for finding 

 -» the centre of gravity of a plane quadrilateral, which may 

 be described as follows. 



Let the intersection of the two diagonals (say Q) be called the 

 cross- centre ; the intersection of the lines bisecting the middle 

 points of pairs of opposite sides (say 0) the mid-centre (which, 

 it may be observed, is the centre of gravity of the four angles 

 viewed as equal weights) ; then the centre of gravity is in the 

 line joining these two centres produced past the latter (the mid- 

 centre), and at a distance from it equal to one- third of the 

 distance between the two centres ; in a word, if G be the centre 

 of gravity of the quadrilateral, QOG will be in a right line, and 

 0G=4Q0. 



The frustum of a pyramid is the nearest analogue in space to 

 a quadrilateral in piano, since the latter may be regarded as the 

 frustum of a triangle. The analogy, however, is not perfect, in- 

 asmuch as a quadrilateral may be regarded as a frustum of either 

 of two triangles, but the pyramid to which a given frustum belongs 

 is determinate. Hence a priori reasonable doubts might have been 

 entertained as to the possibility of extending to the pyramidal 

 frustum the geometrical method of centering the plane quadri- 

 lateral. The investigation subjoined dispels this doubt, and 

 will be found to lead to the perfect satisfaction, under a some- 

 what unexpected form, of the hoped for analogy. 



Let abc, ufty be the two triangular faces, au, b/3, cy the edges 

 of the quadrilateral faces of a pyramidal frustum. Then this frus- 

 tum may be resolved in six different ways into the sum total of 

 three pyramids, as shown in the annexed double triad of schemes. 



* Communicated by the Author. 



