Ta / T \x-1
M ~ --- ( ----------- ).
g \g[sigma]a^2/
Since x is undetermined, all that we can conclude is that M is of the
form
Ta / T \
M = ---.F( ----------- ), (1)
g \g[sigma]a^2/
where F denotes an arbitrary function.
Dynamical similarity requires that T/g[sigma]a^2 be constant; or, if g
be supposed to be so, that a^2 varies as T/[sigma]. If this condition be
satisfied, the mass (or weight) of the drop is proportional to T and to
a.
If Tate's law be true, that _ceteris paribus_ M varies as a, it follows
from (1) that F is constant. For all fluids and for all similar tubes
similarly wetted, the weight of a drop would then be proportional not
only to the diameter of the tube, but also to the superficial tension,
and it would be independent of the density.
Careful observations with special precautions to ensure the cleanliness
of the water have shown that over a considerable range, the departure
from Tate's law is not great. The results give material for the
determination of the function F in (1).
+----------------+------------+
| T/(9[sigma]a^2)| gM/Ta |
+----------------+------------+
| 2.58 | 4.13 |
| 1.16 | 3.97 |
| 0.708 | 3.80 |
| 0.441 | 3.73 |
| 0.277 | 3.78 |
| 0.220 | 3.90 |
| 0.169 | 4.06 |
+----------------+------------+
In the preceding table, applicable to thin-walled tubes, the first
column gives the values of T/g[sigma]a^2, and the second column those of
gM/Ta, all the quantities concerned being in C.G.S. measure, or other
consistent system. From this the weight of a drop of any liquid of which
the density and surface tension are known, can be calculated. For many
purposes it may suffice to treat F as a constant, say 3.8. The formula
for the weight of a drop is then simply