344 G. H. KNIBBS. 
these observati ged,whether according to the categories, 
to the values of R, or of n, they fail to disclose any very striking 
indication of a regular variation. Hence there seems to be no 
alternative, but to base the law, as has been done, upon the mean 
results at the bottom of Table G. 
It may be noticed that St. Venant puts m=1, in his 1850 paper, 
while Hagen in his experiments in 1853 obtained 1-25, agreeing, 
it will be observed, with Reynolds’ definition, m= 3-1; 1bey 
3— 1°75. 
17. An empirical generalization of the indices g and m con- 
sidered as varying both with n and R.—Returning to the question 
of possible variations in the indices g and m, it has already: 
been noticed that gq seems to diminish with the increase of 
internal agitation—in other words when n is large—and also, 
that there is a suspicion that m diminishes as # increases. First 
in regard tog. Any attempt to generalize the value of qg, must 
take account of the fact that when n=1, g=1. If Hagen’s, 
Mair’s, and Unwin’s experiments be relied upon, the values.of 7 
determined by them must be systematically included. And if, a8 
I feel convinced is the case, g is always positive, it must not vanish 
for any value of n. This may be effected by putting it in some 
_ such form as 
a ce (35) 
a(n-1)*+¢a sd tia 
in which if « = 0°18, a = 1, and z = 2, the results will be as 
shewn in§13. So that the es eka function is at least roughly 
Flag = & oe (36) 
and this is true for either régime. : 
It is obvious that when n=1, g=1; when n=2 q is ‘15, andg 
does not become zero till m=. When accurate values of % 
determined for tubes with different values of n, are obtained, there 
will be no difficulty in adjusting (35) to them, by suitably choosing 
z,aandz. The results by the formula are shewn in Table EB. $13, _ 
for comparison with Hagen’s, Mair’s, and Unwin’s observations. — 
