149.

the question

Let SP, and the traced by it, then by

Let = 2, 4, 6...... 2n successively, then the corresponding areas, described by 1, 2, 3, ...... n revolutions of p are evi

Р

150.

The equation to the conchoid referred to the centre

C (Fig. 65,) of revolution of its generating line p, is

See Appendix to new edition of Simpson, p. 335, or Lacroix.

b

b

A BCQ = × BQ = = √(p—a)3 — b3 =

/

2

and since A BCQ, PQN are similar

▲ PQN ▲ BCQ ×

a2

(p—a) 2

This may also be expressed in terms of PN y, or BN = x.

a result which is the same as that deduced by Simpson in his Fluxions, p. 147, from the equation

x2y2 = (b + y)o × (a2 — y2.)

151.

Let d, d', be the diagonals of the trapezium, & their inclination, m, m', the segments of d'. Then, since the trapezium

sum of two A, whose common base is d =

152. Let x, y, be the co-ordinates of the hyperbola Vv, (Fig. 33,) originating in the centre A, and measured along the asymptotes, then by property of the curve

xy const. = m, the equation.

Let BC touch the curve in P, then the subtangent MC

Hence PB PC, and ▲ ACB = 2 AMPN = 2xy × sin. A =2m sin. A a const. Q.E.D.

A geometrical demonstration may be seen in most books on conic sections.

153. Let the radius of the circle CA (Fig. 66,) = 7, ANx, PNy; then by similar A

154.

Let A, B, C, be angles of the A, of which make B = 0, A-30, the side AB = a, and BC; then

156.

Let a, b, C, denote the two sides of the ▲ plane or spherical, and C the included; then in the former case, the

Now in the latter, A, B, being the other two, it is well

known that the area

=A+B+ C − «.

..the spherical A, T is given by the equation

C

2

2

which the reader will have no difficulty in investigating.

157. Let ABC (Fig. 67,) be either end of the oblique triangular prism A'C, and (supposing the prism to be produced,) let Cbc be a section made by a plane its axis, and to its sides; then NCD being the intersection of the planes ACB, bcC, and D that of the lines bc, ND, if through Ab, Bc we make planes to pass

ND, the bNA, cMB between the intersections of these planes with those of ACB, bCc, will each be the inclination of the plane ABC to the plane bCc, i. e., the complement of the inclination of ABC to the axis, (since it is, by construction, to bCc). Let