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a

In this case, AE is a circular arc, whose equation is que = ax – x, or y= Vax - **.

a - 2.3

2.7 The fuxion of this gives y =

*; 2 var x2 2y

4ax + 4.22 a- 4y2 hence j2 = 4.ga

**; consequently

4yo a's?

ai * + y = and ź = VŽE + j? =

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4y2

2y

2y'

This value of ż, the fluxion of a circular arc, may be found more easily thus: In the fig. to art. 60, the two triangles EDC, Eae are equiangular, being each of them equiangular to the triangle ETC; conseq. ED : EC :: Ea : Ee, that is,

ax y:a::*:<= the same as before.

The value of į being found, by substitution is obtained 2cyż = acă for the fluxion of the spherical surface, generated by the circular arc in revolving about the diameter AD. And the fluent of this gives acx for the said surface of the spherical segment BAE.

But ac is equal to the whole circumference of the generating circle ; and therefore it follows, that the surface of any spherical segment, is equal to the same circumference of the generating circle, drawn into x or AD, the height of the segment.

Also when c or ad becomes equal to the whole diameter a, the expression acx becomes aca or ca’, or 4 times the area of the generating circle, for the surface of the whole sphere,

And these agree with the rules before found in Mensuration of Solids.

EXAM. 2. To find the surface of a spheroid.
Exam. 3. To find the surface of a paraboloid.
Exam. 4. To find the surface of an hyperboloid.

TO FIND THE CONTENTS OF SOLIDS.

66. ANY solid which is formed by the revolution of a curve about its axis (see last fig.), may also be conceived to be generated by the motion of the plane of an expanding circle, moving perpendicularly along the axis. And therefore the area of that circle being drawn into the fuxion of the axis, will produce the flution of the solid. That is, AD X area of the circle BCF, whose radius is DE, or diameter BE, is the fluxion of the solid, by art. 9.

fore Note,

67. Hence, if ad = x, DE = y,c= 3.1416; because cy? is equal to the area of the circle BCF; therefore cyer is the fluxion of the solid. Consequently if, from the given equa. tion of the curve, the value of either y or z be found, and that value substituted for it in the expression cy is, the fluent of the resulting quantity, being taken, will be the solidity of the figure proposed.

EXAMPLES.

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Exam. 1. To find the solidity of a sphere, or any segment. The equation to the generating circle being y = ax where a denotes the diameter, by substitution, the general fluxion of the solid cy'i, becomes caxi – cx'x, the fluent of which gives caxi – jcx, or fcx (3a – 2x), for the solid content of the spherical segment Bae, whose height ad is x.

When the segment becomes equal to the whole sphere, then x = a, and the above expression for the solidity, becomes ca’ for the solid content of the whole sphere.

And these deductions agree with the rules before given and demonstrated in the Mensuration of Solids.

EXAM. 2. To find the solidity of a spheroid.
Exam. 3. To find the solidity of a paraboloid.
EXAM. 4. To find the solidity of an hyperboloid.

TO FIND LOGARITHMS.

68. It has been proved, art. 23, that the fluxion of the hyperbolic logarithm of a quantity, is equal to the fluxion of the quantity divided by the same quantity. Therefore, when any quantity is proposed, to find its logarithm ; take the fluxion of that quantity, and divide it by the same quantity; then take the fluent of the quotient, either in a series or otherwise, and it will be the logarithm sought; when corrected as usual, if need be ; that is, the hyperbolic logarithm.

69. But, for any other logarithm, multiply the hyperbolic logarithm, above found, by the modulus of the system, for the logarithm sought.

a+!.

a

a

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of the log.

+

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a

x2

Note. The modulus of the hyperbolic logarithms, is 1; and the modulus of the common logarithms, is 43429448190 &c; and, in general, the modulus of any system, is equal to the logarithm of 10 in that system divided by the number 2.3025850929940&c, which is the hyp. log. of 10. Also, the hyp. log. of any number, is in proportion to the com. log. of the same number, as unity or 1 is to o43429&c, or as the number 2.302585&c, is to l; and therefore, if the common log. of any number be multiplied by 2.302585&c, it will give the hyp. log. of the same number; or if the hyp. log. be divided by 2.302585&c, or multiplied by •43429&c, it will give the common logarithm.

x Exam. 1. To find the log. of

Denoting any proposed number %, whose logarithm is required to be found, by the compound expression a + x

* the fluxion of the number 2, is and the fluxion * X 3

23 jc

+ &c. a t *

a? a3 04 Then the fluent of these terms give the logarithm of a + x

2013 xt or logarithm of

&c.
2a? 3a3

4a
x2 x 3

314 Writing - x for x, gives log.

2a2 393 - 4a*

&c. Div. these numb. and a + x 2.7

2.73

2x5 subtr. their logs. gives Flog.

+ &c.

303 bas

atr Also, because

afx :15 or log.

i afx

atx

x? 2013 therefore log. of is

+ a+2

2a?

4a4 x2 23

X4 and the log. of

+ +
2a

3a3
a

x2 the prod. gives log a?

+ &c. Now, for an example in numbers, suppose it were required to compute the common logarithm of the number 2. This will be best done by the series, atx

x2

x7 log. of

2m x 0
+

&c.
За 5a5 7a?

+

4

a

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a

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+

a

a

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a

40+

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.72

= 2, gives a = 3.r; conseq. * = $, and

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at x
Making
= 5, which is the constant factor for every succeeding term;
also, 2m = 2 X •43429448190 = .868588964; therefore the
calculation will be conveniently made, by first dividing this
number by 3, then the quotients successively by 9, and
lastly these quotients in order by the respective numbers
1, 3, 5, 7, 9, &c, and after that, adding all the terms toge-
ther, as follows:

3 ) •868588964
9 ) 289529654 1) •289529654 •239529654
9 ) 32169962 3 ) 32169962 ( 10723321
9 ) 3574440 5). 3574440

714888 9 ) 397160 7) 397160

56737 9 ) 44129 9 ) 44129 (

4903 9 4903 ! 11 )

4903 (

446 9) 545 13

545 (

42 9)

61
15 )

61 (

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Sum of the terms gives log. 2 = •301029995

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TO FIND THE POINTS OF INFLEXION, OR OF

CONTRARY FLEXURE IN CURVES 70. The Point of

T

E Inflexion in a curve, is that point of it which separates the concave from the convex part, lying between thetwo; or where the curve

A changes from concave to convex, or from convex to concave, on the såme side of the curve. Such as the point e in the annexed figures, where the former of the two is concave

towards

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or —

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towards the axis Ad, from A to e, but convex from E to F; and on the contrary, the latter figure is convex from A to E, and concave from E to F.

71. From the nature of curvature, as has been remarked before at art. 28, it is evident, that when a curve is concave towards an axis, then the fluxion of the ordinate decreases, or is in a decreasing ratio, with regard to the fluxion of the absciss; but, on the contrary, that it increases, or is in an increasing ratio to the fluxion of the absciss, when the curve is convex towards the axis; and consequently those two fluxions are in a constant ratio at the point of inflexion, where the curve is neither convex nor concave; that is, * is.

ä to y in a constant ratio, or is a constant quantity. But constant quantities have no fluxion, or their fluxion is equal to nothing ; so that in this case, the fluxion of j or of is equal to nothing. And hence we have this

Ý general rule: 72. Put the given equation of the curve into fluxions ;

j ä from which find either

Then take the fluxion of

j this ratio, or fraction, and put it equal to 0 or nothing; and

j from this last equation find also the value of the same Then put this latter value equal to the former, which will form an equation; from which, and the first given equation of the curve, x and y will be determined, being the absciss and ordinate answering to the point of inflexion in the curve, as required.

or

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.

or

x

EXAMPLES.

Exam. 1. To find the point of inflexion in the curve whose equation is ar? = a'y + xy. This equation in fluxions is 2axx = a'y + 2.xyx + x*y

a’j , a? +.22

Then the fluxion of this quantity y made = 0, gives 2.ré (ar—ry)=(a? +x+)* (axi - xy-.xy);

: x

a2 + x2 and this again gives

j
a? x2

g

which gives ;=2ax – 2xy

o

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х

Lastly, this value of

being put equal the former, gives

0% + 22

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