140

COMPARISON OF CURVES.

TRACT 31.

be that of n to 1, viz, ac : ac :: n : 1, or ac vz. n x ac, orz = ny at that place ; therefore then also is nax -f- nbx z +ncx 1 &c = a rx + Br z x z + cr 3 .r 3 &c ; hence, by equating thelike terms of this equation, we have na = a?’, nb — b>%nc — cr 3 , See ; and since these are all constant quantities,these equations will express their constant or general rela-tion: let now na, nb, nc, &c, be substituted instead oftheir values a r, b r 2 , o 3 , &c, in the general series express-ing the value of z, and it will be always

z = nax + nbx z -f ncx 1 &c ;but it is always - y — ax + bx 1 -f- cx l &c ;therefore z = ny, or.z :y :: n : 1, i. e. eg : eg :: a c : ac.

Again, when ac is once — ac, or n= 1, then alwaysz =y, or it is always eg = eg. q. e. d.

PROPOSITION II.

The two curves of the same kind being still related as specifiedin the last proposition: then, the corresponding areas aegc,a egc, or adc, Adc, or edg, edg, will always be equal toeach other, or in the same constant ratio with the ordinates.

Demon. If the corresponding ordinates be supposed toflow from any position, always parallel to themselves: then,since the fluxion of the area is equal to the ordinate drawninto the fluxion of the abscissa; and, the fluxions of theabscissas are equal, or in a constant ratio, the fluxions ofthe areas will be in a constant ratio also ; but when twofluxions are in a constant ratio, their fluents are likewise inthe same constant ratio ; therefore the areas generated areas the generating ordinates; but the ordinates are eitherequal or in a constant ratio; therefore the areas are alsoequal or in the same constant ratio, q. e. d.