by Means of the Measures of Angles and Ratios,

The fame otherwise.

Let (if possible) y = AM^ x flu. zAM - /'* 4 BM? Z x flu.zAM-i* -f- CM"* x flu. zAM~ r * &c. (A, B, C, &c. /, q, r,&c. being constant quantities to be determined) : then, bytaking the fluxions, we shall have

4-= pAM px xflu.zA M~t z -j- A A -j- yBM> z xflu. zA M~ ? z -(- &c.A—p'AM^x st. zA M-^ -\-p A A -f ? 2 BM? z x fl.irA M-? z &c.

•^=/ 5 AM pic xflu.zAM - ‘ x -\-p l Aa-\-^~- —&c.

A — /> 4 AM?* xflu. zAM~ fK 4-/> 3 A A 4- 4-^ 4- 44 £? s .

*r 1 * 1 1 z 1 zz 1 zzz

Which values being substituted in the given equation, and thehomologous terms being compared, we shall thereby get^> 4 -{-^ 3

4 bp' 4~ cp 4 d — o, q* 4- a f 4" If 4* cc l 4“ d — °> & c '

also \ P l±ft±^±l x A + q' + aq' -Ybq-\-c X B 1 _ ^lr 3 ar 1 brc x C 4" * 3 4^* 4 ^ 4 c X D j^‘4^>4^ X A4-0 4- aq-fbY. B 4~ r'+ar ~\~b X C 4~ 3 4~^ 3 4”^ X D—op-\-axA-\-q-\-ay.B-\-r-\-a'X.C-\-s-\-a'x.'D — oA4-B + C4-D = o. ‘

Now, from the former of these equations, p* 4- ap* 4 bp l4* cp 4“ d — o, q+ 4 * *zy 3 4" bq 1, -j- cq d — o, &c. it ap-pears evident, that />, y, r, Sec. are the roots of the equationX 4 4 axi 4 bx* A- ex -\- d — o (or, more generally, ofx n 4 ax n ~’-\~ bx n ~* &c. = o, n denoting the order to whichthe fluxions ascend in the given equation); which roots beingtherefore found (by any of the known methods) the values ofp, q, r, &c. will be obtained. But to find from thence andthe remaining equations, the values of A, B, C, ©V. let thelast of these equations multiplied by a, be subtracted from thepreceding one, so shall paqBrCsD — o moreover,let this new equation multiplied by a, be subtracted from thelast but two, and from the remainder let bA 4 bB 4 bC 4-/>D = o be again subtracted, whence will be had p l A 4 - y*B