b=16, and z, found in the fecond example to

rule 1, 36.

Then prh√1+% = 2513*27412287 × 10583 =2659797 the furface nearly.

RULE VI.

For Both Spheroids.

This rule will be the fame as the 6th rule to the last problem, ufing z inftead of q, and b for f, as in the laft rule. That is,

PX (A — B) = {prb × [√] ± }z — ÷ (12)] the furface nearly.

EXAMPLE I.

Taking the ift example to the laft rule; we have P = 1884-955592, A = √1 −÷2 = 97816157, and B = X (I — 2) = 43484444.

Then PX (AB) = 1884-955592 × 97797 18434298 the furface nearly.

EXAMPLE II.

Taking here the 2d example to the laft rule; we have Pprb 2513.27412287, A = √1+% =10583005, and в = × (1+ ≈) = ·47—•

6

Then PX(A-B)=2513*27412287 × 10569409 2656.3821 the furface nearly.

RULE

RULE VII.

For Both Spheroids.

This rule alfo will be the fame with the 7th to the last problem, ufing z and b instead of q and f.

9

That is, 7PX (ABC) = 47 prb ×

c) 27

= the furface very nearly.

EXAMPLE I.

Taking, ftill, the fame example of the oblong fpheroid; we shall have, as before, P = 1884-955592, A = 97816157, and B = 43484444; alfo

I

c = -1 × (1 - 12 — 12x) = 28977188.

8 27

Let there be taken, again, the oblate fruftum, in which, as before, P = 2513 27412287, A = 10583005, and в = 47; alfo

=

27

2x) = 31311407.

Then 27PX(A-B-C)=2513 27412287×105714 2656.8825 the furface very nearly.

the furface

SCHOLIU M.

It is evident that the double of the fruftum will give the middle zone; and that the fruftum being added to, or taken from, half the fpheroid, will give the greater or lefs fegment.

PROBLEM XI.

To find the Solidity of a Spheroid.

RULE I.

Multiply continually together the fixed axe, the fquare of the revolving axe, and the number 52359877 or of 3.14159, and the last product will be the folidity.*

That

*

DEMONSTRATION.

CM

E

Put f B1 the fixed femiaxe, r Iм the revolving femi-axe of the spheroid, a = SI any femi-diameter of the fection NBM, b = IK its femi-conjugate, y = AE an ordinate to the diameter SI, or a femi-axe of the elliptic fection AFC parallel to KL, and z = EF its other femi-axe, alfo = EI, 5 =

B

A

K

N

the fine of the angle AES, or of the angle KIS, to the radius 1,

and p

= 3*14159.

Then, by the property of the ellipfe KSL, aa: bb:: aa- xx:

But the fluxion of the folid KACL is psyxy by writing

abs its value rf; and hence the flucnt frrx X

3aa

xx

pfrr x

aaa

will be the value of the fruftum KACL;

which, when EI or becomes sI or a, gives pfrr for the value

of

of the femi-fpheroid KSI.; or the whole fpheroid = FRR, putting F and R for the whole fixed and revolving axes. 2.E.D.

Corol. 1. From the foregoing demonftration it appears that the value of the general fruftum KAECL is expreffed by 3aa x x

pfrrx x

-

aaa

And if for fr be fubftituted its value abs, the fame fruftum

will alfo be expreffed by pbrsx X

-xx

3 aa

aa

which, by writing ≈ instead of its value, gives ¡psx × (2br+yz) for the value of the fruftum, viz. The fum of the area of the lefs end and twice that of the greater, drawn into one-third of the altitude or distance of the ends.

And out of this laft expreffion may be expunged any one of the four quantities b, r, y, z, by means of the proportion br::y:2.

When the ends of the fruftum are perpendicular to the fixed axe; then a is f, and the value of the fruftum becomes 3ff - xx prr x x for the value of the fruftum whofe ends - ff..

are perpendicular to the fixed axe, its altitude being x.

And when the ends of the fruftum are parallel to the fixed axe, a is = r, and the expreffion for fuch a fruftum becomes 3rr xx pfx x

Corol. 2. If to or from 3pfrr, the value of the femi-fpheroid, be added or fubtracted frrx x

Заа -xx

the value of the

· general

RULE II.

2

Multiply the area of the generating ellipfe, by of the revolving axe, and the product will be the content of the fpheroid..

That is, A the oblate, and CA the oblong fpheroid; where A is the area of the ellipfe.

And this rule is evidently taken from the former.

E X

general fruftum KACL, there will refult pfrrbb x

3a-b

aaa

for

the value of a general fegment, either greater or less than the femi-fpheroid, whofe height, taken upon the diameter paffing through its vertex and center of its base, is ha±x. When a coincides with f, the above expreffion becomes 3f- b prrbb × for the value of a fegment whose base is ff

perpendicular to the fixed axe-And here if we put R for the radius of the fegment's bafe, and for rr its value

RRff

2fbbb the faid fegment will become рRRь ×

3f-b

2f-b

And when a coincides with r, the general expreffion will become pfbb x 3r-b

r

for the value of the fegment whose base is parallel to the fixed axe.-And if we put F, R, for the two femi-axes, of the elliptic bafe of this fegment, refpectively correfponding or parallel to f, r, the femi-axes of the generating ellipfe, when parallel to the base of the segment,

f

and for and r fubstitute their values

r

the faid fruftum will be expreffed by

in which the dimenfions of itself only are concerned.

་

+bb

2 R

Corol. 3. A femi-fpheroid is equal to of a cylinder, or to double a cone, of the fame bafe and height; or they are in proportion as the numbers 3, 2, 1. For the cylinder is 4nfrr nfrr, the femi-fpheroid infrr, and the cone frr.

Corol. 4. When fr, the fpheroid becomes a sphere, and the expreflion furr for the femi-fpheroid becomes nr for the

2