302 THE PRINCIPLE OF SIMILITUDE [392
where F denotes an arbitrary function. Experiment shows that F varies but little and that within somewhat wide limits it may be taken to be 3'8. Within these limits Tate's law that M varies as a holds good.
In the ^Eolian harp, if we may put out of account the compressibility and the viscosity of the air, the pitch (n) is a function of the velocity of the wind (v) and the diameter (d) of the wire. It then follows from similarity that the pitch is directly as v and inversely as d, as was found experimentally by Strouhal. If we include viscosity (v), the form is
n = v/d.f(v/vd), where/ is arbitrary.
As a last example let us consider, somewhat in detail, Boussinesq's problem of the steady passage of heat from a good conductor immersed in a stream of fluid moving (at a distance from the solid) with velocity v. The fluid is treated as incompressible and for the present as in viscid, while the solid has always the same shape and presentation to the stream. In these circumstances the total heat (/?/) passing in unit time is a function of the linear dimension of the solid (a), the temperature-difference (0), the stream-velocity (v}, the capacity for heat of the fluid per unit volume (c), and the conductivity («.). The density of the fluid clearly does not enter into the question. We have now to consider the "dimensions" of the various symbols.
Those of a are (Length)1,
t „ „ v „ (Length)1 (Time)-1,
„ „ 6 „ (Temperature)1,
,,M „ c „ (Heat)1 (Length)"3 (Temp.)-1,
„ „ K „ (Heat)1 (Length)-1 (Temp.)-1 (Time)-1,
„ „ h „ (Pleat)1 (Time)-1.
Hence if we assume
h = axdVv*cuKvt we have
by heat l=.u + v,
by temperature 0 = y — u — v, by length Q = X + z - 3u - v,
by time - 1 = — z — v ;
, .j h = Kad
/avcV i — . \ K J
Since x is undetermined, any number of terms of this form may be combined, and all that we can conclude is thation of viscosity shows that the critical velocity is inversely proportional to the diameter of the sphere.