( ?44 ) 
expreft by the Powers of x\ you mud then transform 
the Equation by this Propofition. Sir l faac Neirton and 
Mr. de Moivre do this by the Reverfion of Seriefes; but 
I take this to be the more proper and more genuine Me- 
thod of doing it diredly. In the fourth and fifth Propo- 
rtions are explain’d the Method of judging of the Na- 
ture and Number of the Conditions that may accompany 
an Incremental or a Fluxional Equation. This is a Cir- 
cumftance that l don’t find to have been explain’d by 
any one before, and the Propofitions are fomething in- 
tricate; wherefore it will not be improper to explain this 
Matter a little more at large. Suppofe then that * and at 
are two variable Quantities, and fuppofing a to increafe 
uniformly by the continual Addition of itsconflant Incre- 
ment (according to the Notation I make ufe of in this 
Book) fuppofe * -f z, — Then if it be propofed as a 
Problem to find the Value of at, exprefs’d by the Powers 
of z, and quite freed from the Increments j by the fourth 
Propofition there may be three Conditions added to this 
Problem. The Demonflration of this is taken from the 
Formation of the Integrals by a continual Addition of 
their Increments. Suppofe that all the Values of z>, x f 
x t x , &C, were fee down in order in fo many Co- 
lumns, and that at the Head of the Table, the corres- 
ponding Values of z, x, *, *, *, were expreft by the Sym- 
bols a, c, c, c, S Then by means of the given Equati- 
on * 4 - z ~ x, or * = a: — z, will C : =c — 4 , whence the 
fecond correfponding Values of x t *, x , * will be a 
e-\- c , c.-f* c t c-\.c— aC = c ■?) and c 4“ f — 4— z. 
(by the Eq.) Whence again the third Values are4 -{- 2 z, 
f _J . - zc -f- c, c 2 c -f- c — 4 ( = c "{~ 2 c Hf “ f. ) 
c T z e — 2, 4+c — z, t and c +2 c q-c — 4 — 2 
and fo you may proceed to find the fourth, fifth, and all 
the. 
