Dimensions of Units

It may be added that if we had assumed a force (or accelera tion) proportional to the inverse Ph power of the distance the index of R would be instead of 3.

Indeterminateness.

In the examples given not more than three indices are required to be determined. In most cases, how ever, the number is greater than three. It is obvious that there are only three dimensional equations depending respectively on the indices of mass, length and time. Hence any additional indices must remain undetermined. A complete mechanical treatment of the problem (if possible) would show that all these values are definite and would determine them. Since however the real utility of the method of dimensions is in con nection with those problems, which, owing to the complexity of the mathematical relations cannot be solved in detail, it follows that for the determination of all but three of the unknown quantities it is necessary to have recourse to experiment. The possibility of solving the equation even for only three variables is however an immense advantage since it reduces the range of experiments necessary for the determination of the rest.

Motion of Fluids.

Casual examination of the flow of liquids in such as can be made by watching the motion of solid particles carried down with the stream, is sufficient to show that it is different in character for sufficiently slow and for sufficiently high rates of flow. Careful investigation demonstrates that for a circular pipe of definite diameter there is a particular outflow at (or very near) which the change occurs. The change is detected by varying the pressure-head and finding at each stage the cor responding outflow; for low heads the two quantities are propor tional to one another; at very high velocities the head varies more nearly as the square of the outflow, The difference is B.

where and h indicate different functions. Each of the terms in brackets is a numeric.

These equations represent the most general conclusions that can be drawn, in this problem, from the method of dimensions. Since each side is a pure number its value is an absolute con stant, i.e., it is independent of the particular system of units in terms of which the values of v, D, p, etc., are expressed.

though of course if mixed systems are employed the value will be different; for example if v is in feet per second, D must not be in inches without a suitable change being made in the "con stant." Such mixed units will not be further referred to though they are frequently employed). This forms the basis of the use of a model. The expression vDp/ p. can be determined experi mentally by means of a single experiment and must be constant for dynamical similarity if it be assumed that the above specifica tion of the problem is complete. It is useful to enquire at this stage in which particulars it may be incomplete. It has already been restricted to cylindrical tubes. A truly cylindrical tube is an ideal one; real tubes depart from the ideal in possessing irregularities of surface. The degree of roughness may be spec ified by the maximum deviations, d, from a perfectly smooth surface, in which case the ratio d/D is another numerical fac tor on which vDp/A will depend; a function of it requires to be introduced on the right. and the single experiment will only determine the numerical value of vDp/ 1.1. for those cases in which the roughness bears a constant proportion to the diameter. Similar remarks require to be made if the tubes are smooth but not cylindrical. If they are elliptical in section a function of the eccentricity will be concerned; this will, however, be a mere number; hence, provided, the eccentricity is the same in the group of cases considered, the " constant " will not be affected. Further, the question of the possible influence of the elastic properties of the fluid must in any case be raised; in practice it is found that they are of no importance until veloc ities comparable with the velocity of sound in the fluid are set up. These questions are raised here in order to indicate that some degree of caution is neces sary; but since, in any case, the results arrived at are intended to be checked by suitable experi ments no irreparable harm is done if some important variable is at first left out of account.