y6 The Resolution of certain fluxionary Equations y

r C — o: and, in the same manner, from the first

equation, will be had pi A qi B r 5 C -4* r’D — i (be-

cause i, and not o, forms the latter part of that equation).

Now, from each of the equations (A B -f- C -)- D = o,pA -j- yB -s- rC 4- jD = o, p r A 4- y*B 4" ^ 2 C 4" — o,

pi A 4* y 3 B 4" r3 C "1“ • j3 B> = I) thus derived, let the preced-ing one multiplied- by p, be subtracted : so shall

q — p . B 4 ~ r — p . C s — p . D — o,

q — p .. yB -}~ r — p - rC -s- r — p . sD — o,

q — p . y B 4 ~ r — p . r 1 C 4" s — p • = i.

Moreover, from each of these last equations, let the preced-ing one multiplied by q, be in like manner subtracted; whencewill be had

r — p .r — q. C 4~— P' s — y - O — o,

r — p .r — q .rC Ar s — P • s — y . rF> — I.

Again, from the last of these, let the preceding one multi-plied by r, be subtracted ; then will s — p . s — q ,s — r . D

= i, and consequently D . . . : whence it is

s — p.s — q.s — r

manifest by inspection (because p is the same with respect to A,

as s is to D, &c.) that A — - -—-, B *= - 1 -,

p — i-P —r.&c. q — p.q —r.&c-

C — ■ 1 —-, &c. From whence the value of y (—

r —p . r — q . &c.

AM /K xflu.zAM — ^-{“BM^xflu.iiAM - ?*6cc.) will be known,let the orders of fluxions in the equation ascend to what heightthey will.—Thus, for example, let the equation propounded

be A 4- ftf-y — M 7rz .* in which cafe, a being = o, b = w‘,

c — o, &c. our general equation, x n 4- A~ bx n ~’ 1 - 4- cx’~i££c. = o, will therefore become x* -\- in =. o ; whereof thetwo roots (p and q) are ms/ —i, and — m \/— i ; from

whence A (== —h-) =- -— , and B (— ——) — —

p — 2?n</—i K 9—p'

I

2mV — i

also, because A is here = M 77 *, we have y = AM^

x