20

Libr. I. Sect. I.

VI. sin i v.s/r — sia i F y/-

P

(e — i) cos FP

V (e—i)uSimili modo addendo vtrimque i fit

„ . (e - 4 -1) b

an| F\/ -77—

v cos F

. p . (e-\-i)b

= Hu— = i(«—

u

A~Il. cos \ v.sf r = cos \F\J-

(e+1) cosi*

— = cos 2 F\/

= §(w+i)v/'

— i- (« +1) v/'

(g—1) &

COS jp'

(e—1 )b

(e + 1) u “ w

Diuidendo VI per VII ad III reueniremus; multiplicatio producit

VIII. r sin v = pcolaxigip tangF = b tmgyj tangi 1 '

= i p cotang yj (u—~) = £ b tangy* (u — “)

E combinatione aequatt. II, V porro facile deducitur

1 j ®

IX. rcosv~b(e — ~ o&F ) =i b ( 2 e — u — —)

X. r

22 .

Per differentiationem formulae IV prodit (spectando y/ vt quantitatem con-

stantem)

rr dv =

d u

u

~ i ('

tang i(p + yj) — tang i {? ■

■yi)j d v =z

r tangy/

d v ; hinc

pr

u tang 7,

'P

d u, siue substituendo pro r valorem ex X,

rr&v

bb tang 7^/

(

hu

-)

v)

dzj

Integrando deinde ita, vt integrale in perihelio euanescat, fit

J

frrdv — bbtangyj (i e (u — —) — logu) = ht\/p. y/(i +^) r= lttangyj\/b.\/{1

Logarithmus hic est hyperbolicus • quodsi logarithmos e systemate Briggico vel ge-neraliter e systemate cuius modulus — X adhibere placet, massaque p (quam procorpore in hyperbola incedente haud determinabilem esse supponere possumus) ne-giigitur, aequatio hancce formam induit:uu — 1 211

XI. iXe

u

logu