Page:Somerville Mechanism of the heavens.djvu/101

Thus if u be given in terms of x, y, z; the four quantities λ, X', Y', and Z', will be determined. If the condition of constraint expressed by u = 0 be pressure against a surface, $\scriptstyle {{\text{R}}_{'}}$ is the re-action.

Thus the general equation of a particle of matter moving on a curved surface, or subject to any given condition of constraint, is proved to be

$\scriptstyle 0=\left({{\text{X}}-{\frac {{d^{2}}{x}}{d{t^{2}}}}}\right)\delta {x}+\left({{\text{Y}}-{\frac {{d^{2}}{y}}{d{t^{2}}}}}\right)\delta {y}+\left({{\text{Z}}-{\frac {{d^{2}}{z}}{d{t^{2}}}}}\right)\delta {z}$ + λδu (10).

70. The whole theory of the motion of a particle of matter is contained in equations (6) and (10); but the finite values of these equations can only be found when the variations of the forces are expressed at least implicitly in functions of the distance of the moving particle from their origin.

71. When the particle is free, if the forces X, Y, Z, be eliminated

from

$\scriptstyle {\text{X}}-{\frac {{d^{2}}{x}}{d{t^{2}}}}=0;\ {\text{Y}}-{\frac {{d^{2}}{y}}{d{t^{2}}}}=0;\ {\text{Z}}-{\frac {{d^{2}}{z}}{d{t^{2}}}}=0$

by functions of the distance, these equations, which then may be integrated at least by approximation, will only contain space and time; and by the elimination of the latter, two equations will remain, both functions of the co-ordinates which will determine the curve in which the particle moves.

72. Because the force which urges a particle of matter in motion, is given in functions of the indefinitely small increments of the co-ordinates, the path or trajectory of the particle depends on the nature of the force. Hence if the force be given, the curve in which the particle moves may be found; and if the curve be given, the law of the force may be determined.

73. Since one constant quantity may vanish from an equation at each differentiation, so one must be added at each integration; hence the integral of the three equations of the motion of a particle being of the second order, will contain six arbitrary constant quantities, which are the data of the problem, and are determined in each case either by observation, or by some known circumstances peculiar to each problem.

74. In most cases finite values of the general equation of the motion of a particle cannot be obtained, unless the law according to which the force varies with the distance be known; but by assuming from experience, that the intensity of the forces in nature