Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/26

increments ${\ddot {p}}',{\ddot {x}}',{\ddot {y}}',{\ddot {z}}'$, and subtracting the original value of the expression from the value thus modified. Now,

${\begin{aligned}{\mathfrak {S}}[P({\ddot {p}}+{\ddot {p}}')]&-\textstyle \sum \displaystyle [{\tfrac {1}{2}}m{({\ddot {x}}+{\ddot {x}}')^{2}+({\ddot {y}}+{\ddot {y}}')^{2}+({\ddot {z}}+{\ddot {z}}')^{2}}]\\&-{\mathfrak {S}}(P{\ddot {p}})+\textstyle \sum \displaystyle [{\tfrac {1}{2}}m({\ddot {x}}^{2}+{\ddot {y}}^{2}+{\ddot {z}}^{2})]\\&={\mathfrak {S}}(P{\ddot {p}}')-\textstyle \sum \displaystyle [m({\ddot {x}}{\ddot {x}}'+{\ddot {y}}{\ddot {y}}'+{\ddot {z}}{\ddot {z}}')]-\textstyle \sum \displaystyle [{\tfrac {1}{2}}m({\ddot {x}}'^{2}+{\ddot {y}}'^{2}+{\ddot {z}}'^{2})].\end{aligned}}$

But since ${\ddot {p}}',{\ddot {x}}',{\ddot {y}}',{\ddot {z}}'$ are proportional to and of the same sign with possible values of $\delta {\ddot {p}}',\delta {\ddot {x}}',\delta {\ddot {y}}',\delta {\ddot {z}}'$, we have, by the general formula of motion,

${\mathfrak {S}}(P{\ddot {p}}')-\textstyle \sum \displaystyle [m({\ddot {x}}{\ddot {x}}'+{\ddot {y}}{\ddot {y}}'+{\ddot {z}}{\ddot {z}}')]\leqq 0.$

The second member of the preceding equation is therefore negative. The first member is therefore negative, which proves the proposition with respect to (15). The demonstration is precisely the same with respect to (13) and (14), which may be regarded as particular cases of (15).

To show the same with regard to (16) and (17), we have only to observe that the quantities affected by $\delta$ in these formulae differ from those affected by the same symbol in (14) and (15) only by the terms

$\textstyle \sum \displaystyle ({\dot {X}}{\dot {x}}+{\dot {Y}}{\dot {y}}+{\dot {Z}}{\dot {z}})$and${\mathfrak {S}}({\dot {P}}{\dot {p}}),$

which will not be affected by any change in the accelerations of the system.

When the forces are determined by the configuration (with or without the time), the principle may be enunciated as follows: The accelerations in the system are always such that the acceleration of the rate of work done by the forces diminished by one-half the sum of the products of the masses of the particles by the squares of their accelerations has the greatest possible value.

The formula (17), although in appearance less simple than (15), not only is more easily enunciated in words, but has the advantage that the quantity ${\frac {d}{dt}}{\mathfrak {S}}(P{\dot {p}})$ is entirely determined by the system with its forces and motions, which is not the case with ${\mathfrak {S}}(P{\ddot {p}})$. The value of the latter expression depends upon the manner in which we choose to represent the forces. For example, if a material point is revolving in a circle under the influence of a central force, we may write either $X{\ddot {x}}+Y{\ddot {y}}+Z{\ddot {z}}$ or $R{\ddot {r}}$ for $P{\ddot {p}}$, $R$ and $r$ denoting respectively the force and radius vector. Now $X{\ddot {x}}+Y{\ddot {y}}+Z{\ddot {z}}$ is manifestly unequal to $R{\ddot {r}}$. But $X{\dot {x}}+Y{\dot {y}}+Z{\dot {z}}$ is equal to $R{\ddot {r}}$, and ${\frac {d}{dt}}(X{\dot {x}}+Y{\dot {y}}+Z{\dot {z}})$ is equal to ${\frac {d}{dt}}(R{\dot {r}})$.

It may not be without interest to see what shape our general formulæ will take in one of the most important cases of forces dependent upon the velocities. If a body which can be treated as a point is moving in a medium which presents a resistance expressed by