EXAMPLE.

Required the convex furfaces of the two hoofs of a conic fruftum, whose base and top diameters are 30 and 19, and height 18 inches; the fection being made through the contrary extremities of the diameters: together with that of the oblique cone ACV.

D+d

A

B

P= √ Dd = 243√30 × 19 = 243 × 24

2

x segment of the circle AB whofe height is DX

d-AD

X (D2 x tabular fegment whofe height is

{( D +d) -
d-

- AD

AD

is d—AD).

Alfo

I. QX PX 785398 = 2*73326528 × 590'4 = 1613.7198213 inches 11.2063876 feet, the convex furface of the oblique cone ACV.

2. x 2 × 785398 × (D2-P) = 785398 × 3*48010217 X (900 - 5904) = 2.73326528 × 309.6846.2189307 inches = 5.87652 feet, that of the hoof ABC.

3. ex 785398 × (Pd2) = *785398 × 3.48010217 X (509 4-368.64)=2*73326528 X 221.76 6061289085 inches = 42092285 feet, that of the complemental hoof ACG.

PRO

Alfo that of the complemental hoof in cor. 2, will become

fegment of the circle AB whofe height is AB X

D-AD

d-AD

=

DB

X

GCAD

GCAD

=

GC

× feg. of the circle AB whofe height is D x dAD

'4b2 + (D − d) 2
D-d

d

X (−nd2 + D2 x tab. feg. whofe height is

Corol. 4.

If D

coincide with A, the rules in the last corollary

X (AB2 X N- 1 X AB X/AI

face of the ungula ABC.

PROBLEM XXX.

To find the Convex Surfaces of the Elliptic Ungulas of a Cone, made by a Plane cutting off a Part of the Bafe.

Multiply continually together the four following quantities, viz. the quotient arifing from the divifion

of

GC

X AIAB X GC X n = 222 X VB X AIV =

AB

Dd for that of the oblique cone

X (AIAB X GC—GC2)= =

n4b2+(D-d)2 D- d

Dd-d2) for that of the complemental elliptic

Corol. 5. If DC be parallel to AV, or the fection a parabola; fince its area B is DC X DF = DC AD X DB, the general theor. for the ungula will become

[feg. FBE to height DB (D-d) √d (D-d)] for the con

vex surface of the parabolic ungula FEBC.

And the rule in cor. I will be

VB

× (feg. FAE + DI√AD XDB) =

√4b2 + (D d)

OB

D-d

[(feg. FAE to height AE + (D−d) √d (D − d)] for that of the part AEFCV.

AD + (D − d) √d (v — d) — ndd), for that of the comple

mental parabolic ungula FAECG.

Corol.

of the fquare of the lefs diameter by that of the greater; the quotient arifing from the divifion of the difference between the part of the diameter of the bafe not cut off and half the fum of the diameters, by the difference between the faid part of the diameter of the bafe and the lefs diameter; the fquare root of the quotient arifing from the divifion of the part cut off the diameter of the bafe, by the difference between the part not cut off and the lefs diameter; and fuch a fegment of the base of the fruftum, whofe height is equal to the product arifing from the multiplica

tion

Corol. 6. If the angle CDB be greater than the angle VAB, or the fection be an hyperbola, its area being found, and substituted for B in the general rules, will give the furfaces of the hyperbolic ungulas.

Corol. 7. If the hyperbolic fection be perpendicular to the bafe, DI will vanish, and the general theorems will become CB × feg. FBE x feg. FBE its height being IB =

VB

OB

=

IB

> feg. of the circle AB, whofe height is

for the curve furface of the perpendicular ungula CIB.

VB

D

2

Alfo × fegment FAE = × fegment whofe height is

OB

CB

IB

x feg. of the cir. AB whofe height is

for that of the remaining part AICV.

of the circle AB whofe height is

complemental perpendicular ungula A1CG.

tion of the quotient of the greater diameter divided by the lefs, into the difference between the lefs diameter and the part of the base diameter not cut off : and call the laft product R.-Then

1. The difference between R and the base of the hoof, multiplied by the quantity Q, in the laft problem, will give the convex furface of the ungula required.

V

2. The fum of R and the remaining fegment of the base, not included by the hoof above, multiplied by, will give the furface of the remaining part of the whole cone.

3. From the furface of the part in the last article, fubtract that of the little cone at the top, viz. the product of o by the area of the top of the fruftum ; and the remainder will be the furface of the complemental ungula.

R

That