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101. 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 fluxion of the axis, will produce the fluxion 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.
102. Hence, if AD-X, DE—y, c=3·1416; because cy2 is equal to the area of the circle BCF: therefore cy is the fluxion of the solid. Consequently if, from the given equation of the curve, the value of either y2 or x be found, and that value substituted for it in the expression cy2x, the fluent of the resulting quantity, being taken, will be the solidity of the figure proposed.
EXAM. 1. To find the solidity of a sphere, or any segment. The equation to the generating circle being y2 = ax - x2, where a denotes the diameter, by substitution, the general fluxion of the solid cy2x, becomes caxx - cx2x, the fluent of which gives cax2-cx3, or cx2 (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 xa, and the above expression for the solidity, becomes ¿ca3 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.
TO FIND LOGARITHMS.
108. Ir has been proved, art 23, that the fluxion of the byperbolic 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.
104. But. for any other logarithm, multiply the hyperbolic logarithm, above found, by the modulus of the system, for the logarithm sought.
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 43429 &c. or as the number 2.302585 &c. is to 1; 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.
of a +x
EXAM. 1. To find the log. of
Denoting any proposed number z, whose logarithm is required to be found, by the compound expression
a+, the fluxion of the number z, is
these terms give the logarithm of z or
a 2α2 3a3-4a4
α 2a2 Sa3 4α4 a+x· 2x 2x3 2x5 + + &c. α X α 3a35a5
the prod. gives log. a2x2
a + x
+ -of- + &c.
a2 2a4 3a6
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,
which is the constant factor for every succeeding term; also, 2m=2×·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 together, as follows:
Sum of the terms gives log. 2·301029995
EXAM. 2. To find the log. of "+
TO FIND THE POINTS OF INFLEXION, OR OF CON-
105. THE Point of Inflexion in a curve, is
that point of it which separates the concave from the convex part, lying between the two;
or where the curve
changes from concave to convex, or from convex to concave, on the same side of the curve. Such as the point E in the annexed figures, where the former of the two is concave towards the axis AD, from A to E, but covex from E to F; and on the contrary, the latter figure is convex from A to E, and concave from E to F.
106. 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 fluxion of the ab-
quantities have no fluxion, or their fluxion is equal to nothing;
thing. And hence we have this general rule:
107. Put the given equation of the curve into fluxions ;,
from which find either or
ratio, or fraction, and put it equal to O or nothing; and from
this last equation find also the value of the same
Then put this latter value equal to the former, which will
EXAM. 1. To find the point of inflexion in the curve whose equation is ax2 =α2y + x2y,
This equation in fluxions is 2axx = a3y + 2xyx + x2ÿ
Then the fluxion of this quantity
a2 + x2 x
and this again gives ==
Lastly, this value of being put equal the former, gives
=1a, the ordinate of the point of inflexion sought.
EXAM. 2. To find the point of inflexion in a curve defined
by the equation aya√ ax + x2.
EXAM. 3. To find the point of inflexion in a curve defined by the equation ay2 a2x + x3.
EXAM. 4. To find the point of inflexion in the Conchoid of Nicomedes, which is gene,
rated or constructed in this manner: From
a fixed point P, which is called the pole of FG
distances, FA, GB, HC, IE, &c. equal to each other, and equal to a given line; then the curve line ABCE &c. will be the cons choid; a curve so called by its inventor Nicomedes.
TO FIND THE RADIUS OF CURVATURE OF CURVES,
108. THE Curvature of a Circle is constant, or the same in every point of it, and its radius is the radius of curvature. But the case is different in other curves, every one of which has its curvature continually varying, either increasing or decreasing, and every point having a degree of curvature peculiar to itself; and the radius of a circle which has the same curvature with the curve at any given point, is the radius of curvature at that point; which radius it is the business of this chapter to find.
109. Let Are be any curve, concave towards its axis AD; draw an ordinate De to the point E, where the curvature is to be found: and suppose Ec perpendicular to the curve, and equal to the radius of curvature sought, or equal to the radius of a circle having the same curvature there, and with that radius describe the said equally curved circle BEе; lastly, draw Ed parallel to AD, and de parallel and indefinitely near to DE: thereby making Ed the fluxion or increment of the absciss ad, also de the fluxion of the ordinate DE, and Ee that of the curve AE. Then put x
AD, Y➡DE, Z—AE, and r≈CE the radius of curvature; then is Ed=x, de=y, and Eez,
Now, by sim. triangles, the three lines Ed, de, Ee,
are respectively as the three therefore
or x, y, z,
x + Gc. x =GE. y + GEÿ
x - x
and the flux. of this eq. is GC or, because GC —— BG, it is GC BG. v=GE ÿ +GE. y. But since the two curves AE and BE have the same curva ture at the point E, their abscisses and ordinates have the same Auxions