The general rule, therefore, for finding points of inflection may be thus enunciated. First ascertain the values of x and y which will satisfy simultaneously the equations

F= 0, Vu - 2 UVw + Uv = 0,

and reject all the pairs of values thus obtained which do not, as we pass through the corresponding points, correspond to a change of sign in the expression

Vu - 2 UVw + U2v,

or which do correspond to a change of sign in either U or V: the pairs of values of x and y, which are retained, correspond to points of inflection.

Secondly, ascertain those pairs of values which satisfy simultaneously

F= 0, U=0, V = 0,

and reject all of these pairs which do not correspond to a change of sign in both U and V as we pass through the corresponding points along one or other of the branches, or which do correspond to a change of sign in the expression

In the preceding investigation we have supposed F to be a rational function of x and y. Should this not be the case it will be evident, from what has been said, that in addition to the values of x and y, which may be obtained by the rule which we have enunciated, we must likewise take those which will render in the first case,

the conditions depending on change of sign being the same as before.

Hence, from the formula (10) there is, if we cast out constant

factors,

xy (ax3 + by3) = 0 ;

or, by the equation to the curve,

Thus x = 0, or y

=

0, and in both cases neither U nor V changes sign, while the formula (9) does change sign. Hence we have two points of inflection,

and suppose that we wish to find whether there be a point of inflection at the origin. Then

U = 2x (2x2 + 2y3 — a3),

V = 2y (2x2 + 2y3 + b2),

u = 12x2 + 4y2 - 2a2,

v = 4x2 + 12y2 + 262,

W = 8xy.

F

From these results it is evident that U and V both change sign if we change x and y each of them from 0 to 0. Moreover it is clear that neither Vu, Uv, nor 2UVw, experience any change of sign when we put ±x, ±y, for x, y, respectively. Hence the expression (9) does not change sign. If we had kept y positive or negative throughout, while we changed x from 0 to 0, the expression (9) would have changed sign, and flexure would not have taken place. Hence we see that the branch which passes through the origin from below to above the axis of x, or that which passes from above to below, will have an inflection at the origin.

=

Putting this expression =∞, we get x = 0, or y 0; and therefore, by the equation to the curve, y = b, x = a, respectively. It is evident, then, that as x passes through 0, U and V do not change sign while the expression (9) does; and similarly for y: hence there are two points of inflection, viz. x = 0, y = b, and x = a, y = 0.

CHAPTER V.

ON THE INDEX OF CURVATURE, THE RADIUS OF CURVATURE, AND THE CENTRE OF CURVATURE, OF A PLANE CURVE.

128. LET,

Index of Curvature.

+ dy, be the inclinations of the tangents PT, QS, at points P, Q, of a curve AB, very near to each

other, to the axis of x. Then it is evident that the greater be the angle dy between the two tangents, for a given small arc PQ, or the smaller be the arc PQ for an assigned value of the small angle dy, the greater will be the curvature of the curve in the vicinity of P. Hence, proceeding to the limit, when the arc PQ or 8s becomes less than any assignable mag

nitude, we see that is a measure of the curvature at P.

d↓ ds

df

The angled is called the angle of contingence, and the ds index of curvature at P.

Radius and Centre of Curvature.

129. From P and Q draw two normals PC, QC, meeting in C. These two normals will evidently include an angle 84.

α

Let PC = p, QC = p', the chord of PQ = c, and let 1⁄2π denote the angle between the chord of PQ and the normal QC. Then it is plain that

but, proceeding to the limit, when dy, c, and a, become less than any assignable quantities, sin dy and c vanish in a ratio of equality with d and ds respectively: hence

Now, p and p' being ultimately in a ratio of equality, it follows that a circle described with C as a centre, and touching the line PT in P, will ultimately touch the line QS in Q. The angle of contingence will accordingly be the same ultimately for the circle as for the curve: also the arc PQ in the circle will be ultimately in a ratio of equality with the arc PQ in the curve, since each of these arcs, by the 7th Lemma of Newton's Principia, vanishes in a ratio of equality with their common chord. Hence a circle so described has the same curvature as the curve at the point P. This circle is called the osculating circle, or the circle of curvature at the point P, p the radius, and C the centre of curvature.

The equation (1) shews that the index of curvature at any point of a curve is equal to the reciprocal of the radius of the osculating circle.

Expression for p when x is the Independent Variable. 130. Differentiating the equation (Art. 100)