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If u = o be the equation to the curve, the following expression for the radius of curvature is frequently convenient, viz.
du du du du (du dul?
or, if u consist of the sum of two parts, the one involving a alone and the other y alone,
ď u du dul
(1) In the parabola, the equation to which is
ye = 4mx,
4 (m + x)
(3) In the rectangular hyperbola referred to its asymptotes
XY = m?,
(r? + y)
(4) In all the curves of the second order the radius of curvature varies as the cube of the normal.
If N be the length of the normal, N2 =
All the curves of the second order are included in the equation
y' = 2 px +qx®;
dy y 3 9
-p?. d x2
(a + c)3
(6) In the semi-cubical parabola 3 ay = 2x",
(2a + 3x)3 x
= 0. d x
(9) In the tractory y + (a? – yo)? Taking the expression for p in which y is the independent variable we find,
(a’ – y').
(10) In the hypocycloid xi + ył = a), p = 9 (a xy)).
If the curve be referred to polar co-ordinates q and 0, then
or, if it be expressed by the relation between r and the perpendicular on the tangent (p),
(12) In the lemniscate of Bernoulli me = a' cos 20,
a (15) The equation to the lituus being po?
ro ( 4 a' + gut)! p:
2a*(4 a' – 904) (16) The equation to the trisectrix being r=a(2 cos 0+1),
(5 + 4 cos 0)|
3 (3 + 2 cos 6) (17) In the logarithmic spiral when referred to p and r,
р p = mr,
(18) In the involute of the circle p' = po- a, and p = p.
br (19) The equation to Cotes' spirals is p =
(a + gol)! r (a® + po?)? P р
a' b (20) In the epicycloid
CP (pu2 - a')
c - a? p= p
(co – a’)(po– aʼ)!
SECT. 2. Evolutes of Curves. When a curve is referred to rectanglar co-ordinates, the co-ordinates (a, b) of its centre of curvature are given by the equations
a = X
or, if u = o be the equation to the curve,
+ d x dy dy d x2 då dy dx dy dx) dy ?
To determine the equation to the evolute it is necessary to eliminate « and y between these equations and that of the given curve; but the complication of the formulæ renders this elimination always very troublesome, and most frequently impracticable. The few cases in which it can be effected we shall give. (1) In the parabola y = 4a x, whence
y? a = 3x + 2a, B
Substituting these values in the equation to the parabola,
or 27aß = 4 (a - 2a), the equation to the semi-cubical parabola.
(2) In the rectangular hyperbola referred to its asymp
whence 2a = 30 + 2ß = 3y +
y3 + 2003 Yi + 3x+ y + 3y+ x + r.3 2 (a + b) = 3 (x + y) +
ma or 2m2 (a + B) = (x + y)”, or x + y = (2 m2)} (a + B). Similarly, subtracting
2 m2 (a - b) = (x - y)", or x - y = (2m?)! (a - ).