and therefore, from (2), we have for the equation to the envelop

which is the equation to an ellipse similar in form to either of the original ones, its axes being of twice the magnitude.

Again, if λ = 1, we have from (3),

which are the equations to a point, viz. the centre of the fixed ellipse, through which it is evident that all the moveable ellipses pass.

Intersection of Consecutive Normals to a Curve.

147. The equation to the normal at any point x, y, of a curve ƒ (x', y') = 0, will be

where u = f(x, y). Differentiating (1), considering æ as a vari

dx' dy

From (1) and (2), and the differential of the equation ¿ =

0,

the values of x', y', the coordinates of the intersection of two consecutive normals, coinciding with their values obtained otherwise in Art. (146).

CHAPTER IX.

DIFFERENTIALS OF AREAS, VOLUMES, ARCS, AND SURFACES.

Differential of an Area.

148. Let AB be any portion of a curve referred to rectangular axes Ox, Oy. Let PM, QN, be the ordinates of two points P, Q, of the curve, very near to each other. Let AC be the

ordinates of A. Draw PR and QS, at right and MP produced. Let OM = x, PM = y,

angles to QN ON = x + dx, the area ACMP, and A + SA = the

Then, since the area PQNM is evidently less than SQNM and greater than PRNM, it is plain that

Proceeding to the limit, when dx and Sy become less than any assignable magnitudes, y + dy = y: hence ultimately, replacing small increments by differentials, we see that

Differential of a Volume of Revolution.

149. Conceive a surface to be generated by the complete revolution of the curve AB, in the diagram of the preceding Article, about the axis of x. Let V represent the volume generated by the area ACMP, and V+ SV that generated by the area ACNQ. Now it is shewn in ordinary treatises on Trigonometry that the area of a circle is equal to T. (radius), where is the circular measure of 180°: hence the areas of the circles generated by the revolution of the ordinates PM, QN, will be equal to

пу?, п(у+ бу),

and therefore the volumes of the thin cylinders generated by the revolution of the areas PRNM, SQNM, will be equal to

πηδα, π (y + δυ) δα.

But it is plain that SV is greater than the former and less than the latter of these cylinders: hence

Now ultimately, when dx and dy become less than any assignable magnitudes, πу' and π (y+dy), the limits of the value of SV

δε

become equal to each other: hence, replacing indefinitely

small increments by differentials, we have

150. Let the chord of the arc PQ in the figure of Art. (148) be denoted by c; let arc AP :

=

s, arc PQ Ss. Then

=

Now ultimately, when 8x is diminished without limit, we know by the 7th Lemma of Newton's Principia that 1: ds

C

hence, replacing infinitesimal differences by differentials, we have

a relation which has been already established in Art. (100).

Differential of a Surface of Revolution.

151. Let S, S+ SS, denote the areas of the surfaces generated by the revolution of the arcs AP, AQ, of the curve AB, about

to the axis of x and each equal to the arc PQ.

Let APs, PQ = ds. Then, if Pp, Qq, revolve about Ox together with the rest of the diagram, they will generate two thin cylinders. The length of each of these cylinders will be equal to the curvilinear distance between the circular ends of the surface generated by PQ; but the average radii of the circular sections of this surface will evidently be greater than those of the former and less than those of the latter cylinder. Hence it is manifest that, 2ryds and 2 (y+dy) ds being the surfaces of the two cylinders,