Hence, the osculating plane at any point contains the resultant of the impressed forces.

Again, if p be the radius of absolute curvature,

which expresses that the portion of the pressure due to the velocity is equal to that produced by the impressed forces.

190. If the terminal points are not definitely assigned (if, for instance, it be required to find the line of swiftest descent from one given curve to another) we have no longer

Sx=0, Sy=0, Sz = 0

at the limits; but, with the requisite modifications, the process in § 189 enables us to find the proper conditions in any These questions, however, belong rather to Calculus of Variations than to Kinetics.

case.

Thus, suppose that the final point of the path is to lie on

δε

(1).

Also that [] may vanish, which is necessary in order that may be zero, we must have

dt

dx dy dz
Sx+ Sy +
dt

Sz =0

dt

...........(2).

Now the only relation between Sx, dy and Sz is (1), to which (2) must therefore be equivalent: hence

These equations show that the moving particle meets the terminal surface at right angles. A similar condition is easily seen to hold if the initial point of the path is also to lie on a given surface, provided the whole energy be given and the given surface be an equipotential one. If it be not equipotential, terms depending on dx,, Sy, Sz,, will appear in the integral and must be taken along with {}.

If a terminal point is to lie in a given curve the condition is to be determined in a similar manner.

191. A particle moves under the action of given forces on a given smooth surface; to determine the motion, and the pressure on the surface.

Let

F(x, y, z) =0

..(1),

be the equation to the surface, R the force acting in the normal to the surface, which is the only effect of the constraint. Then if λ, μ, v be its direction cosines, we know that

with similar expressions for μ and v; the differential coefficients being partial.

If X, Y, Z be the impressed forces, our equations of motion are, evidently,

R disappears from this equation, for its coefficient is

and vanishes, because the line whose direction cosines are pro

dx portional to &c. being the tangent to the path, is perdt'

pendicular to the normal to the surface.

If we suppose X, Y, Z to be such forces as occur in nature, (Chap. II.) the integral of (4) will be of the form,

and the velocity at any point will depend only on the initial circumstances of projection, and not on the form of the path pursued.

To find R, multiply equations (3) in order by λ, u, v, add, and observe that x2+μ22=1. We thus obtain

2

d2x d'y d2z

λ +μ +v = Xλ + Yμ + Zv + R.
dť2 dt2

dť2

But, if p be the radius of curvature of the normal section through Ss, P1 the radius of absolute curvature of the path, we have, by Meunier's Theorem,

which gives the normal pressure on the surface.

192. To find the curve which the particle describes on the surface.

For this purpose we must eliminate R from equations (3). The result is

two equations, between which if t be eliminated, the result is the differential equation to a second surface intersecting the first in the curve described.

193. So far for the general case, let us now make particular hypotheses.

If there be no impressed forces on the particle, we have by (5), v2 = C, and equations (6) become, since in this case

d2x

Now &c. are proportional to the direction cosines of 9 ds the radius of absolute curvature of the path; λ, μ, v are those of the normal to the surface. Hence those lines coincide, or the normal to the surface lies in the osculating plane to the path.

But this is the property of the longest or shortest line joining two points on a surface, hence we have the following,

If a particle, subject to no forces, move from one point to another of a smooth surface, the length of the path described

will be a maximum or minimum.

This result will be afterwards deduced from a different principle (Chap. IX.).

194. A particle moves on a surface of revolution, the only force acting being gravity parallel to the axis of the surface; to determine the motion.

Take the axis of the surface as that of z, the equation may be written

F(x, y, z) =ƒ{√(x2 + y2)} − z = 0.

This may be put in the form

f(p) - z = 0,

if p be the distance of any point in the surface from the axis.