Marcu 18, 1915] 
NATURE 
67 

Ina gaseous medium, of which the particles 
repel one another with a force inversely as the 
nth power of the distance, the viscosity is as the 
{n+3)/(2n—2) power of the absolute temperature. 
Thus, if m=5, the viscosity is proportional to 
temperature. . 
Eiffel found that the resistance to a sphere 
moving through air changes its character some- 
what suddenly at a certain velocity. The con- 
sideration of viscosity shows that the critical 
velocity is inversely proportional to the diameter 
or the sphere. 
If viscosity may be neglected, the mass (M) of 
a drop of liquid, delivered slowly from a tube of 
diameter (a), depends further upon (T) the capil- 
lary tension, the density (7), and the acceleration 
of gravity (g). If these data suffice, it follows 
from similarity that 
M=22p( E ), 
where F denotes an arbitrary function. Experi- 
ment shows that I varies but little and that within 
somewhat wide limits may be taken to be 38. 
Within these limits Tate’s law that M varies as 
a holds good. 
In the AZolian harp, if we may put out of 
account the compressibility and the viscosity of the 
air, the pitch (m) 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 vis- 
cosity (v), the form is 
n=v/af (v/vd), 
where f is arbitrary. 
As a last example let us consider, somewhat in 
detail, Boussinesq’s problem of the steady pas- 
sage 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 inviscid, 
while the solid has always the same shape and 
presentation to the stream. In these circum- 
stances the total heat (2) passing in unit time is 
a function of the linear dimension of the solid 
(a), the temperature-difference (@), the stream- 
velocity (v), the capacity for heat of the fluid 
per unit volume (c), and the conductivity (k). 
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 (Leneth)}, 
once: ic aw.) (Lensth)! (Lime) =3, 

eee, 6 (Temperature)!, 
Rass) sens c¢.. (Heat)! (Length)—3 (Temp.)—1, 
SORES k (Heat)! (Length)! (Temp.)~! (Time) ~}, 
Sure = h (Heat)! (Time)~!. 
Hence if we assume 
h=a® Ov? oR", 
we have 
by heat I=u+v 
by temperature o=y—w—v, 
by length o=41*+2—3u-—7, 
by time —I=-f-U; 
NO. 2368, VOL. 95 | 


so that 
h=xad (=). 
K 
Since ¢ is undetermined, any number of terms 
of this form may be combined, and all that we 
can conclude is that 
h=xad.F¥ (avc/k), 
where IF is an arbitrary function of the one 
variable avc/«. An important particular case 
arises when the solid takes the form of a cylin- 
drical wire of any section, the length of which is 
perpendicular to the stream. In strictness similarity 
requires that the length 1 be proportional to the 
linear dimension of the section b; but when 1 is 
relatively very great h must become proportional 
to 1 and a under the functional symbol may be 
replaced by b. Thus 
h=kl6.F (b6vc/k). 
We see that in all cases h is proportional to 
6, and that for a given fluid F is constant pro- 
vided uv be taken inversely as a or b. 
In an important class of cases Boussinesq has 
shown that it is possible to go further and actually 
to determine the form of F. When the layer of fluid 
which receives heat during its passage is very 
thin, the flow of heat is practically in one dimen- 
sion and the circumstances are the same as when 
the plane boundary of a uniform conductor is 
suddenly raised in temperature and so maintained. 
From these considerations it follows that F varies 
as v1, so that in the case of the wire 
hl. /(dvc/K), 
the remaining constant factor being dependent 
upon the shape and purely numerical. But this 
development scarcely belongs to my _ present 
subject. 
It will be remarked that since viscosity is 
neglected, the fluid is regarded as flowing past 
the surface of the solid with finite velocity, a 
serious departure from what happens in practice. 
If we include viscosity in our discussion, the ques- 
tion is of course complicated, but perhaps not so 
much as might be expected. We have merely to 
include another factor, v”, where v is the kine- 
matic viscosity of dimensions (Length)? (Time)-!, 
and we find by the same process as before 
Bree 
ONC ee kK 
Here = and w are both undetermined, and the 
conclusion is that 
hanab FE {OE} 
SIL8 KO) 
where F is an arbitrary function of the two 
variables avc/«k and cv/x. The latter of these, 
being the ratio of the two diffusivities (for 
momentum and for temperature), is of no dimen- 
sions; it appears to be constant for a given kind 
of gas, and to vary only moderately from one gas 
to another. If we may assume the accuracy and 
universality of this law, cv/« is a merely numerical 
constant, the same for all gases, and may be 
