VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. IQS 



flowing moment itself is produced by multiplying the moment of the curve fad, 

 in respect of the axis ph, determined by the foregoing corollary, that is, ca 

 X BD + CB X AD into 4ac ; and therefore it will be 4 ac X ac X bd + 4^ac 

 X CB X AD. Now if this be applied to the librated figure pfadh, or 2ca X 

 AD, by Cor. 5 of this Prop, there will arise the distance of the centre of gravity 



of the figure pfadh from the axis ph, equal to i + -^cb, which is 



equal to half the right line ce, as above determined. 



Corol. 8. — If through the point n, where dt the tangent to the catenaria in 

 D meets the line ak, a right line be drawn parallel to bc, meeting a right line 

 through E parallel to ak in the point o; this point o will be the centre of gravity 

 of the curve ad. For, by Corol. 6, the centre of gravity of the curve ad is in 

 the right line eo. But it will be also demonstrated, that it is in the right line 

 NO, and therefore will be in the point o. Let da be conceived to be librated 

 about the axis hl ; the moment of this is the curve da, drawn into the distance 

 of the centre of gravity from hl, and therefore its fluxion is da X nh, (nh is 

 the fluxion of the distance of the axis of libration, from the centre of gravity, 



which is equal to ^^2ax4-xx X ,. ' = ai'. And therefore the moment 



of the weighty curve da, librated about the axis hl, is ad\ Therefore the dis- 

 tance of the centre of gravity from the same axis is ax applied to ad, or 



^— ^— . But because dt touches the catenaria, by Cor. 4 of this Prop, the 



angle bdt or dny will be equal to agr. And the angles at A and y are right ; 

 therefore in the equiangular triangles rac and dyn, it is ra : AC :: dy : yn. 



Whence yn = , that is, yn is the distance of the centre of gravity of 



the chain ad from the axis hl ; or the said centre is in the right line no. 



Corol. 9. — If upon i a right line be drawn parallel to ar, meeting on pro- 

 duced in w; the point w will be the centre of gravity of the space achd. For, 

 by Corol. 7, this centre is in the right line iw ; and it will be shown presently, 

 that it is in nw, and therefore is the very point w. For in the same manner 

 as in the foregoing, the fluxion of the moment of the space achd, librated 

 about hl, is shown to be achd X hA = ac X ad X hA = ax*</ lax -{■ xx y. 



ax 



, = aax. And therefore the moment of the space achd, librated about 



HL, is equal to the fluent of the fluxion aax, that is to aax. This therefore 

 applied to the space itself achd, or a*^ lax -^ xx, gives the distance of the 

 centre of gravity of the space achd from hl, that is = — rI~* ^"' 



c c 2 



