therefore given, we have two sides, and an angle subtended one of them to find the third side A'C' which will give the inclination required. Thus, by common forms

a' +1'

a + d-- 6



sin. 6'. sin, c' a', b', c', being the sides opposite to A', B', C. Hence by proper reductions and the solution of a quadratic, we get

6 cos. A' sin. c' £ N sin.’a sin. c'sin.' A' tan.

a' é

á + c



2 cos.



or tan.




ZPMC cos. A'. sin. a I v sin." B sin.’a. sin.' A


B + a 2 cos.

COS. 2

2 which, when adapted to logarithmic computation, will give ZPMC or the angle of co-ordinates.

The following process will apply still more generally. It will give us the co-ordinates of the intersection of a plane, and the surface of any solid whatever.

Let A'X', AY', A'Z', (fig. 10. A.) denote the rectangular coordinates of the surface, BC the intersection of the cutting plane with the plane Y'AX, and P be any point in the curve required. Draw PN I plane Y'A'X', and meeting it in N, and NM I BC, and join PM. Also draw MN I AX', and A'A I BC.

Then x', y',z'; x, y, z, denoting the co-ordinates of the surface, and of the intersection respectively, let us suppose, for the greater brevity, that x is measured from A along CB, and y in the plane BPC, I CB. Also put 6 = Z PMN, which = the inclination of the cutting plane with that of x', y'; a = the given line AE, and B = the given angle REM'.

Now PN is parallel to A'Z'.

:: z' = PN = PM x sin. 0 = y. sin. e. Also x = A'E + EM' = a + ER . cos. B = a + (AM RM AE) cos. ß = a + x cos. B - NM x tan. B x cos. ß – a cos. ?B = a. sin. B + x cos. ß - y.cos. 8 sin. B.

In like manner we get

[ocr errors]

y'= NM' = y. cos. 8. cos. B + x. sin. B — a.cos. ß. sin. B.

The three equations
X' = a.sin. *B + x cos. B

y cos. 0. sin. B
ý' = - a . cos. B. sin. B + x sin. B + y .cos. 6. cos. B

z = y. sin. O expressing the co-ordinates of the surface in terms of x and Y,

the given equation of the surface will enable us by elimination to find y in terms of x; i. e. the nature of the section required.

Ex. Let the given surface be that of a right cone, whose angle at the vertex, put = A. Also let its co-ordinates a', y', z', be measured from the vertex, and a coincide with the axis. Then its equation will be

A 2? + y = x' tan.



in which the above values of x', y', z', being substituted, there results an equation of two dimensions. The curve of intersection will, therefore, be an ellipse, circle, hyperbola, or parabola, according to the values of the constants. We leave the discussion of these separate cases, and the application of the above equations to other surfaces, to the reader.


To trace the curve whose equation is y =


1 + x2 Let x = 0, then y = 0, or the curve meets the axis in the origin of co-ordinates A, (fig. 11.)

Let x = t; then y = 0, or AB, Ab are asymptotes to the


To find the maximum or minimum values of y, we have

1 - 22

dx (1 + x?)?

i. x = 1, which may be represented by AM and Am, the abscissæ corresponding to the maxima (PM) and (pm) of y, PM being =

[blocks in formation]
[ocr errors]

Again, since when x = 0 dy

=1= tan. TAM, we have the angle TAM at which the dx curve cuts the axis = 450. To find the points of inflexion

day 2.73 6x

(1 + x2)3
i. r = 0 and r? 3 = 0,

or x = 0, and x = tv3
and the corresponding values of y
y = 0, and y = IN 3



Take, therefore, AM', Am' = 73, 3, and P'M', p’ım' = V3, V3, and the points P', A, p', will be those of contrary

4 flexure. The curve p'AP is concave to the axis, because dạy 2.r3 6.0

is negative between the values of x, dr2 (1 + x2)3 O and 3; and positive, with negative ordinates between 0 and — 3. Hence PD, and p'd, are convex to the axis.

13. To find the equation to the curve of which subnormal : (abscissa)3 :: 2a: 1.

Let y = its ordinate, x its abscissa,

:.x3 :: 2a2 : 1

[blocks in formation]

:: y = ax?, or the curve is the common parabola, the line of abscisse issuing from the vertex at right angles to the axis, and its parameter being

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

14. To find the equation of the curve of which, (ordinate) = 6 x area .

Let the ordinate = y, and abscissa = x.
Then yo = N b. Jydx

:: ya = b sydt
and 443 dy = b ydx
:: 4yody = b dx, and integrating



[ocr errors]

which is a particular case of the cubical parabola.

Fig. 12, will represent the course of this curve, Dd being the line of abscissæ, and A their origin.

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

when I =


which is also the point where the curves cut

3b the axis, or where y = 0. This point is :: C. Tt, the tangent at C, is I axis Dd.

15. Supposing the curve EAF (fig. 13,) to be described by the uniform motion of the point A along CB, and an uniform angular motion round C, it can evidently make no difference whether this angular motion be that of the plane EAC, whilst CB is at rest, or that of CB, in an opposite direction, whilst the plane remains at rest. The curve may, therefore, be conceived as described by the uniform motion of A along the radius vector CA,

which also moves with an uniform angular motion. Hence C will be the pole of the curve; and calling the radius vector Pi

and the < between it, and any radius vector given in position, as CE, 0,

we have

[ocr errors]

i. p = mo, which is the equation of the Spiral of Archimedes.

The applicability of teeth of this form in the construction of pumps, steam engines, and other machines requiring uniform motion in a given direction, is pretty evident.

Thus, supposing F, F, F", three equal and similar teeth made by the union of the equal and similar spirals EF, EʻF; EF, E'F'; E”F, EF, similarly posited around the circumference of the wheel ; then if the piston AB be constrained to move in the vertical direction AB by means of a tube, the uniform motion of the wheel round its axis will cause the piston to move upwards and downwards also with an uniform motion, and the greatest altitude through which the piston will be raised is GF.

16. To construct the spiral whose areas are the measures of the ratios between the ordinates which terminate them, we have d. Area =


=d. log. 2

= M X


(M being the modulus of the sys


tem, and p the radius vector, and e the angle described by it)

2 M de

M :.0 = £

=CF po

[ocr errors]


[ocr errors]

Let p = 00. Then 6 = 0, and C = 0.

M ..0 = F

which is an equation to the Lituus. pa'

(See Cotes' Harmonia Mensurarum, Pars III., p. 85.)

The prime radius vector Ab (Fig. 14,) is evidently an asymptote to the curve.

« ForrigeFortsett »