Principia Mathematica. 52.9

O duarum est dimensionum, id est, termino

m m — m m1 n n

m—rn

OOA~n~

m

m

= 1 * + P Q ? =P * +yAQ +

2f x. T- Jf) ^ 4f ^

•+ &c.

di nata? occurrens in M, ac tandem ordi-nata CG, seu A B*«f- B C~i , elevetur addignitatem cujus est index q atque itä inseriem infinitam convergentem resolvatur,hujus seriei primus terminus erit semper

Lib 1 RPrimu*.

Proh.x t: 1 i i .

1 HIOR.

x 1. vn.

m

Nam sit Radix qu*sita 7 +T> p aequali;seriei infinitae A*\-BZ-\-CZ 2J FDZ*erit a -{-b m aequalis huic seriei addignitate m p e vect*, sumatur ergo seriespotentia: a-\-b m quse erit a "> -f- m a »— 1 b

m x—--4 ■»— * b 1 m X --x

m— z

j 4 ™ — 3 b* & conferantur cum termi-nis dignitatis infinitinomii A+BZ+CZ*+DZ* &c. ad dignitatem p evecti, ( n*. j jo )invenieturque dp=«m; fA?— l BZ

— m 4 m — 2 b; p A p C Z' 1 p X ^ ~ -

Ar-*B 2 Z 2 = mx at-Wi f>Ae- l DZ*

+ f X t 1 Ar — 2 XifiCZJ+pgr r

^ P -3 BJ Z icuix — xm — a b 1

4 " — 3 b i &C.

»t j • - . — tn 4 m 1

Unde invenietur A — a?,BZ ——X -L

P 4 —=

P

ra “P m — 2 p

= ™ a -f h cz 2 = m -*-^--\ P **

p I X i Xp 2 p

&c.

;;2. Lemma. Si in recta A E positio-ne data, ad quam curva Z FH refertur ,capiatur abscissa quivis A B, sitque ordi-nata correspondens F B squalis dignitatiabseist* A B1, in datam quantitatem iductae, & deinde sapiantur intervalla x-qualia BC, CD, & agantur ordinataeC G, D H, ac per punctum F ducatur tan-gens F I ordinati CG occurrens in I, &recta FM parallela lineae A E, eidem or-Jonu l.

B C

aequalis ordinat* FB, insistenti ad initiumquantitatis constantis BC; secundus ter-minus aequalis ei it differentiae inter F B &C I , id est, line* MI, & tertius termi-nus una cum sequentibus in infinitum ae-quabitur line* C I qn* jacet inter tangen-tem & curvam-... . Dem. sit All = ti FB~y, data B C — O, ducta intelligatur or-dinata f b , alteri FB infinite propinquaqu* lineant F M lecet in m, & punctisF, f, coeuntibus erit Frluir, fm — dy,ac triangula F mf , FM 1 similia, ideo-que dx: dyzzO : MI, sed quoniam y — x <1(ex hyp.) & proinde dy~qxH — ' dx,est d x : dy — i : qx c t ~ ergö Ml^zgx 0 . -1 XOLcCl—FB-f-MI — — 'xO>

Pnetereä (ex hyp.) est G C — x-j- O q —+ Q + 9 X q- J xq - 0 2 2<

3 1 X 1

q X q — ix<? — 1 , , , „ ■ • c

——-— - 3 - * q — i 0 * 4 -&c. in rnfi-

1X2x3

ni turn (548). Quare erit GI = GC —C I = t*'*Zj ; , , - X oa + *=?

IX2 IX1X3

X x x .v t

0

o