ELL

371

ELL

C X X 2 rpt c '

y — -=r Iherelore z —

a{a — x % )

; but by

(« 2 - ^

in this last expression

Uutir >g d for — ~ C 'iain

attain * (

--*• for the value of s, But

(« a — x 2 )%

* (a> i ( dx

a 3 )

( " *°)i (a* - “ (a 3 - a?Y X

\* a^i

V throwing the factor ^1 — into an infinite

* es > w „u • • a x i /, d X 1

We obtain z = —-^ x ( 1 — TT-j

<t> ^ (a 2 — a-) \ 2 a 3

a a* x? \ 4 , „ „ ax,

■» e OW&c.j But the fl ^nt of^—^

C ' tc 'e. a -p ^*e corresponding arc, E P, of the circumscribingob t . “erefore, taking a equal to the circular arc, we

^ le fluent of z — a —b - c

2 a 1 2.4a 4

2 .4 .6 a s3 a 3 b— a : 3 (a 3 —a: 3 )

li A — a: (a 3 — nr) 1

_ c —ar 3 _ x ni.

tL ~ 75 --— &c., but when a; becomes = a,

0tll y a s rTf 3 ) 2 = O, consequently the values B, c, d, becomeii_ aS 0 ll °Ws, viz.,

3 a 3 a 3 3 a*

4 x T xa = 27T a

3

3.5a

b v — X -p- X-^- X A

i ese 6 4 2 2.4.6

tL a Ues being substituted in the above series, give

2 . 2 .4.4d

J &c. Now by putting b = ~2~2~’ 0&c. we obtain 2 = A X

= 3.3.5rf 3

2 . 2 . 4 . 4 . 6 . 6 >

—\&c. There*

1 .3 . d 3.5 . d

S lto 31 ~ B -?—

^l 0 ' v in» n ^ circumference of an ellipsis, we have the

%o * "^ u ^iply the circumference of the circumscribing

e by t v d 1 . 3. d

e sum of the infinite series 1 -=■ — b —-

SkO f , & 4

5.7.d 0

b 6 t> ~s 3 — > &c *

kz at nnj °

^etse l~T Be q u i re d the periphery of an ellipsis, the

ax,s of which is 50, and the conjugate 40, then

■-(«)'= ■ -(4)'=> -I——

Therefore 2 ) . 36 = d

2 ) . 18

.09

3

.27

.36 = d

162

81

4 ) . 0972

4 ) . 0243

. 006075 = a 1 -^? - B3 4,4

. 0182255

.091125

.36 = d

546750

273375

6 ) . 03280500

6 ) .00546750

- 3.5 d

.00091125 = B-r—-- 0

5 6> 6

. 004556257

.03189375

.36 = d

19136250

9568125

8 ). 0114817500

8 ) . 00143521875

-- 5.7 d

. 00017940234375 = 0 - 0 - 5 -= »

y 0.0

.00125581640625

9

.01130234765625

.36= d

678140859375G

3390704296875

.0040688451562500