The theory of dimensions often enables us to forecast, to some extent, the manner in which the magnitudes involved in any particular problem will enter into the result. Thus, assuming that the period of a small oscillation of a given pendulum at a given place is a definite quantity, we see that it must vary as √(l/g). For it can only depend on the mass m of the bob, the length l of the string, and the value of g at the place in question; and the above expression is the only combination of these symbols whose dimensions are those of a time, simply. Again, the time of falling from a distance a into a given centre of force varying inversely as the square of the distance will depend only on a and on the constant μ of equation (15). The dimensions of μ/x2 are those of an acceleration; hence the dimensions of μ are L3T−2. Assuming that the time in question varies as axμy, whose dimensions are Lx+3yT−2y, we must have x + 3y = 0, −2y = 1, so that the time of falling will vary as a3/2/√μ, in agreement with (19).
The argument appears in a more demonstrative form in the theory of “similar” systems, or (more precisely) of the similar motion of similar systems. Thus, considering the equations
| d2x | = − | μ | , | d2x′ | = − | μ′ | , |
| dt2 | x2 | dt′2 | x′2 |
(43)
which refer to two particles falling independently into two distinct centres of force, it is obvious that it is possible to have x in a constant ratio to x′, and t in a constant ratio to t′, provided that
| x | : | x′ | = | μ | : | μ′ | , |
| t2 | t′2 | x2 | x′2 |
(44)
and that there is a suitable correspondence between the initial conditions. The relation (44) is equivalent to
| t : t′ = | x3/2 | : | x′3/2 | , |
| μ1/2 | μ′1/2 |
(45)