346 G. H. KNIBBS. 
Hence the observations are not only better represented by making 
m a function of £ itself, but the formula attains to greater gener- 
ality, and empirically expresses the observed results on giving 
that function the form 
7 
Ux Rl ars 7 (38) 
It is easy to see that by suitably choosing x and z, the expression 
(37) may be made to represent a large range of observed values 
of m ascertained by future experiments. At the same time it 
should be noticed that for m to be entirely general, it must remain 
2. as long as m remains 1. Hence the complete expression for m 
must have some such form as 
(37a) 
e+b(n—1)* R* eee 
and the application of this latter or any similar formula might 
perhaps be considered when exact experiments are to hand. 
m=1+ 
It is, very probable I think, that careful experiments will shew 
that m is after all, as implied in this last equation, a function of 
the roughness. In further discussing the solution of the problem 
it is essential to keep in view the fact, that the only simple way 
in which the observed velocities can be accurately represented, 
with the fall in pressure as argument, is by the expression U*= 
k’T, and that the only simple general relation by which all the 
observed velocities can be approximately defined, with the argu- 
ments, radius and fall in pressure, is by the formula 72 k#’R™J, in 
which k” and #’ are the quantities in Table A., and m =1:27 or is 
determined by (37)—(38). But as already remarked in regard to 
the law of variation of the velocity with the radius, the table 
itself presents many anomalous features, and it seems to be 
impossible to accwrately represent all the observations by formule 
of the preceding type, or indeed by any formula whatsoever. The 
real general problem may therefore perhaps be regarded as taking 
the form, not of determining an expression which will absolutely 
reproduce the experimental results—that seems to be impossible— 
but one which shall represent them in their generality, and which 
