Libe. I. Sect. III.

9-G

2 tang (45°+ <y) — cotang(45 °+ «)) 2 = 2 + 4 tang 2 co 2 ; vnde habetur

sini/ 2 tang 2« 2 _ sini/ 2 tang 2 <y 2

^ cos / cos/ ’ cos f cos/

90.

Considerabimus primo casum eum, vbi e solutione aequationis 12 valor non

2g —sin2^

nimis magnus ipsius g emergit, ita vt—-—r——— in seriem secundum potestates

sm g

ipsius sini g progredientem euoluere liceat. Numerator huius expressionis, quamper X denotabimus, iit

= 3 T 2 sin i g 3 — V® sin i g 5 — f sin i g’’ — etc.

Denominator autem

= 8 sin i g 3 —~ 12 sin \ g J + 5 sin i g 7 + etc.

Vnde X. obunet tormam

T.T- - 4

y +1 sili i g 2 + ff sin ig 4 + etc.

Vt autem legem progressionis co efficientium eruamus, differentiamus aequationem

dX

Xsing*~ 2 g- sm 2 g, vnde prodit 5 Xcosg'sing' 2 + sin/—r^—== 2 -2 cos 2 g - 4 sin g 2 ;

Clg"

, • r 2 „ d.V dX

statuendo porro s in?g z =px, fit ~—-—±smg, vnde concluditur—p— —

8—fi.Xcos£- 4 — 5X(i —2 x) dX

-, et proin ( 2 * — 2 xx) 4 - (5 - 6 *) X.

- 6 X cos gsin g 2

Quodsi igitur statuimusX = f ( 1 -f- a x + fixx + y x* + Sx* + etc.)

obtinemus aequationem

—a)xa-f-(5 y —2 {3) x 1 -f (4 S — 5 y) x* -f etc.)=:(8 — 4a) x + (8« — 4 fi) x x+ (8/? — 4y)x 3 + (8y — 4d)a’ 4 + etc.

quae identica esse debet. Hinc colligimus cc = -f-, yzx: 1 ^^ etc.,

vbi lex progressionis obuia est. Habemus itaque

X:

4 , 4.6 . 4.6.S 4.6.8.JO

; t + 3^* T +3—^ XX +

0.5.7 • 9

# 3 +

4.6.3.10.12

3.5.7.9.11

* 4 + etc.