if it acted alone. If the first of these equations be multiplied by ẟx, the second by ẟy, and the third by ẟz, their sum will be
, (6),
and since X - d²xdt²; Y - d²ydt²; Z - d²zdt²; are separately zero, ẟx, ẟy, ẟz, are absolutely arbitrary and independent; and vice versâ, if they are so, this one equation will be equivalent to the three separate ones.
This is the general equation of the motion of a particle of matter, when free to move in every direction.
2nd case.—But if the particle m be not free, it must either be constrained to move on a curve, or on a surface, or be subject to a resistance, or otherwise subject to some condition. But matter is not moved otherwise than by force; therefore, whatever constrains it, or subjects it to conditions, is a force. If a curve, or surface, or a string constrains it, the force is called reaction: if a fluid medium, the force is called resistance; if a condition however abstract, (as for example that it move in a tautochrone,) still this condition, by obliging it to move out of its free course, or with an unnatural velocity, must ultimately resolve itself into force; only that in this case it is an implicit and not an explicit function of the co-ordinates. This new force may therefore be considered first, as involved in X, Y, Z; or secondly, as added to them when it is resolved into X', Y', Z'.
In the first case, if it be regarded as included in X, Y, Z, these really contain an indeterminate function: but the equations
Xdt = d²xdt; Ydt = d²ydt; Zdt = d²zdt,
still subsist; and therefore also equation (6).
Now however, there are not enough of equations to determine x, y, z, in functions of t, because of the unknown forma of X', Y', Z'; but if the equation u = 0, which expresses the condition of restraint, with all its consequences du = 0, ∂u = 0, &c., be superadded to these, there will then be enough to determine the problem. Thus the equations are
u = 0; X - d²xdt² = 0; Y - d²ydt² = 0; Z - d²zdt² = 0.
