194 PHILOSOPHICAL TRANSACTIONS. [aNNO l6g7 . 



Whence the distance of the centre of gravity of the curve fad, from the point 



CAXBD + CBXAD CAXBD , TV, , ^ 



or 4- + -.VCB. Moreover, because of zk pa- 



AD ' ^ 



AR : ZK :: CA : cz; whence cz= , and therefore 



AD ' 



CE, which by construction is equal to ^bc -(- -i^cz, will be equal to i,- 



+ vBC. That is, the centre of gravity of the curve fad, and the point e, de- 

 termined by the construction, will be equally distant from c ; but being in the 

 same right line, and situate the same way, they must necessarily coincide. 



The coincidence of the point e, as above determined, with the centre of 

 equilibrium determined Prop. 5 of this, may be thus shown synthetically. By 

 Corol. 1, Prop. 5, it is 2bax = ayd + ba X ak. Whence ah -j- 2bax = 

 achd -[- ba X ar = (by the foregoing corol.) ar x ca -|- ba x ar. That is, 

 BD X AC + 2bax = ar X CB, Or BD X AC = AR X CB — 2bax, Whence bd 



X AC + AD X BC = ad X BC + AR X CB — 2BAX ^ 2AD X BC — 2BAX = 



2ad X AC + 2AD X AB — 2BAX. And by applying it to 2ad, it will be 



EDXAC , , , ABXAD — BAX , ARX y, ^ ARX . ^, ,. , 



i H iBC = AC -H = CA H . But — IS the distance 



AD AD AR AR 



of the centre of equilibrium of the chain from the vertex a, by Prop. 5, and 

 therefore by the same ca -J is the distance of the point e from c, and 



4r '■ 1- -' Bc is the distance of the same e, from the same c, by this corol. 



- AD ' •' 



Whence it appears that these two determinations of the point e come to the 



, , ARX BDXAC , , 



same, because ca -j := 4- \- -i-BC. 



' ' AR ^ AD ' • 



Corol. 7- — The centre of gravity of the space pfadh is in i, the middle point 

 of the right line ce. For since the centre of gravity of the fluxion of ad, or 

 v>d and f/", is distant as far again from ph, as the centre of gravity of the fluxion 

 of ACHD, or DH/«(/and eypf, and D<i + f/X ac is given, equal to D(//m + p/pp ; 

 it is plain that the centre of gravity e of the fluent fad is as far again distant 

 from PH, as the centre i of the fluent pfadh. But I shall prove this otherwise, 

 after the manner of the foregoing. 



Let an erect cylinder be supposed to be raised upon the figure pfadh, and 

 cut oft' by a plane passing through ph, making half a right angle with the plane 

 of the base ; that solid will represent the moment of the figure pfadh, when 

 poised upon the axis ph. The fluxion of this solid, or of the aforesaid moment, 

 (that is, the solids erected upon vpfp and HDclh) is produced, if the moment of 

 the fluxion, or the fluxion of the moment of ad, is drawn into the given line 

 -J-ac. For by Corol. 3 of this proposition, hdcI/i =: dcI X ac. Wherefore the 



