equation (7), in consequence of the values of X', Y', Z', in the end of article 69, it gives for by article 81 the particle may be considered as free, whence The first member of this equation was shown to be the pressure of the particle on the surface, which thus appears to be equal to the square of the velocity, divided by the radius of curvature. 86. It is evident that when the particle moves on a surface of unequal curvature, the pressure must vary with the radius of cur vature. 87. When the surface is a sphere, the particle will describe that great circle which passes through the primitive direction of its motion. In this case the circle AmB is itself the path of the particle; and in every part of its motion, its pressure on the sphere is equal to the square of the velocity divided by the radius of the circle in which it moves; hence its pressure is constant. 88. Imagine the particle attached to the extremity of a thread assumed to be without mass, whereof the other extremity is fixed to the centre of the surface; it is clear that the pressure which the particle exerts against the circumference is equal to the tension of the thread, provided the particle be restrained in its motion by the thread alone. The effort made by the particle to stretch the thread, in order to get away from the centre, is the centrifugal force. Hence the centrifugal force of a particle revolving about a centre, is equal to the square of its velocity divided by the radius. 89. The plane of the osculating circle, or the plane that passes through two consecutive and indefinitely small sides of the curve described by the particle, is perpendicular to the surface on which the particle moves. And the curve described by the particle is the shortest line that can be drawn between any two points of the surface, consequently this singular law in the motion of a particle on a surface depends on the principle of least action. With regard to the Earth, this curve drawn from point to point on its surface is called a perpendicular to the meridian; such are the lines which have been measured both in France and England, in order to ascertain the true figure of the globe. 90. It appears that when there are no constant or accelerating forces, the pressure of a particle on any point of a curved surface is equal to the square of the velocity divided by the radius of curvature at that point. If to this the pressure due to the accelerating forces be added, the whole pressure of the particle on the surface will be obtained, when the velocity is variable. 91. If the particle moves on a surface, the pressure due to the centrifugal force will be equal to what it would exert against the fig. 24. m curve it describes resolved in the direction of the normal to the surface in that point; that is, it will be equal to the square of the velocity divided by the radius of the osculating circle, and multiplied by the sine of the angle that the plane of that circle makes with the tangent plane to the surface. Let MN, fig. 24, be the path of a particle on the surface; mo the radius of the osculating circle at m, and mD a tangent to the surface at m; then om being radius, oD is the sine of the inclination of the plane of the osculating circle on the plane that is tangent to the surface at m, the centrifugal force is equal to M If to this, the part of the pressure which is due to the accelerat ing forces be added, the sum will be the whole pressure on the surface. 92. It appears that the centrifugal force is that part of the pressure which depends on velocity alone; and when there are no accelerating forces it is the pressure itself. 93. It is very easy to show that in a circle, the centrifugal force is equal and contrary to the central force. Demonstration.-By article 63 a central force F combined with an impulse, causes a particle to describe an indefinitely small arc mA, fig. 25, in the time dt. As the sine may fig. 25. be taken for the tangent, the space described from the impulse alone r being radius. But as the central force causes the particle to move through the space 94. If v and be the velocities of two bodies, moving in circles whose radii are r and r', their velocities are as the circumferences divided by the times of their revolutions; that is, directly as the space, and inversely as the time, since circular motion is uniform. But the radii are as their circumferences, hence Thus the centrifugal forces are as the radii divided by the squares of the times of revolution. 95. With regard to the Earth the times of rotation are everywhere the same; hence the centrifugal forces, in different latitudes, are as the radii of these parallels. These elegant theorems discovered by Huygens, led Newton to the general theory of motion in curves, and to the law of universal gravitation. Motion of Projectiles. 96. From the general equation of motion is also derived the motion of projectiles. Gravitation affords a perpetual example of a continued force; its influence on matter is the same whether at rest or in motion; it penetrates its most intimate recesses, and were it not for the resistance of the air, it would cause all bodies to fall with the same velocity it is exerted at the greatest heights to which man has been able to ascend, and in the most profound depths to which he has penetrated. Its direction is perpendicular to the horizon, and therefore varies for every point on the earth's surface; but in the motion of projectiles it may be assumed to act in parallel straight lines; for, any curves that projectiles could describe on the earth may be esteemed as nothing in comparison of its circumference. The mean radius of the earth is about 4000 miles, and MM. Biot and Gay Lussac ascended in a balloon to the height of about four miles, which is the greatest elevation that has been attained, but even that is only the 1000th part of the radius. The power of gravitation at or near the earth's surface may, without sensible error, be considered as a uniform force; for the decrease of gravitation, inversely as the square of the distance, is hardly perceptible at any height within our reach. 97. Demonstration.-If a particle be projected in a straight line MT, fig. 26, forming any angle whatever with the horizon, it will constantly deviate from the direction MT by the action of the gravitating force, and will describe a curve MN, which is concave towards the horizon, and to which MT is tangent at M. On this particle there Chap. II.] are two forces acting at every instant of its motion: the resistance of the air, which is always in a direction contrary to the motion of the particle; and the force of gravitation, which urges it with an accelerated motion, ac cording to the perpendiculars Ed, Cf, &c. The resistance of the air may be resolved into three partial forces, in the direction of the three axes or, oy, oz, but gravitation acts on the particle in the direction of oz alone. If A represents the resistance of the air, its component force in the axis or is evidently dx A ; for if Am or ds ds be the space proportional to the resistance, then dx Am Ec:: A:A = A ; Ec ds but as this force acts in a direction contrary to the motion of the particle, it must be taken with a negative sign. The resistance in dz ; hence if g be the force of As the particle is free, each of the virtual velocities is zero; hence we have dx dt2 - Ady; dz=g-A dz ; ds' dt ds ds for the determination of the motion of the projectile. If A be eli minated between the two first, it appears that |