8

‘the Nature 'and Inveftigation

The same otherwise.

ix. Let xy be the given Rectangle (as before); andput % ~ x + y, then z 2 being — x* + 2 xy -f- y 2 , we have^z 2 — ’ * 2 — \y % ~ xy. But the Fluxion of {z *—

— (and consequently that of its Equal xy) is zk?— xx—yy (by Art. 6 ): Which, beca use z — xct- y andk=k+y, is also equal to x+yXx+) — xx—yj—yx -f xy.

Corollary i.

12. Hence the Fluxion of the Product of three va-riable Quantities (yzu) may be derived: For, if x beput zu ; then yzu will become ~yx, and its Fluxion— yx + xy (as above :) But x being — zu, and, there-fore, x — 7 . ;k + uk, if these Values be substituted myx

+ xy, it will become j) x ku + iix-f zuy~yzu-\-yuk-{:zny the Fluxion of yzu required. In like Manner theFluxion of xyzu will appear to be xyzu -j- xyzu -+■xyzu + xyzu, and that of xyzuw — xyzuw + xyzuw +xyzuw -j- xyzuw + xyzuw.

Corollary ,2.

u.

' 13. Hence, also, the Fluxion of a Fraction — may

be determined. For, putting x = —, we have xz~ u,

and therefore xk + zx — u (as above) ; whence, by

- . , -r u xk u uz

Transposition and Division, * = — — — --- —^(by

u . . zu—uk .... , u, •

writing — for ar) — - - — ; which is the true rluxi-

0 Z L 2

On of x, or its Equal —, the Fraction proposed.

14. Now, from the foregoing Propositions, andsubsequent Corollaries, the following practical Rules,

for