( 139 )

TRACT XXXI.

COMPARISON OP CURVES.

PROPOSITION I.

If adc, A dc be two curves of any one and the same species,and there be drawn d d a tangent to them both; and from thepoints of contact d, d, there be drawn the diameters dfb, dfb,bisecting the double ordinates eg and ac, eg and a c, whichare parallel to d d : then, if the corresponding ordinates egand eg, or ac and ac , be equal to each other, or in any ratioto each other in any one part, they will always be equal or inthe same constant ratio to each other in every part.

Demon. For, let x denote any abscissa df. Then, sincedf is always to df in a constant ratio, let that ratio be thatof r to 1 ; then will df be denoted by rx. Also let y denotethe double ordinate eg, and z the ordinate eg; and let thegeneral equation y = ax p bx z -f cx 3 + dx* &c, express thegeneral relation of that species of curve, or the relation ofthe ordinate to the abscissa; then will the equation z =A rx + bIx 2 + cr 3 x 3 + Dfb ’ 4 &c, express the general rela-tion between df and eg. Now, in order to determine therelation of the fixed constant coefficients of the terms ofthese two series, let the corresponding ordinates be con-ceived to flow into that situation where they are equal orhave a known ratio, as at ac, ac , and let their ratio there