162 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
and integrating twice 
bo 
1 
b 
~—% 
le (Me Gea Se Sree 1 
ay ine. oy =) dy = — 1325 [= G05 ie hoe i pee EE 
Putting 
vir 
2, for is 
: : 3 4 
the denominator of 5K,, equation (67), becomes 
Cwm 1°325 Labo. 
In a similar manner the numerator is found to be 
bt (F) Ls (ear? + 1:25b,2) + 2L2(10a,2 + 8,2) + 82a,2 + 80,°3, 
and as the coefficients of a, and b, are nearly equal in the numerator, no sensible error 
will be introduced by putting 

by = Chip 
then 
3 Lt + 2 x 5:°53L? + 50 /a\* 
= 0-408L 6) ee) 
which is a minimum if 
Ol i) nis nec) 5 (75) 
and 
Kase BIER te, es eet eee (76). 
Hence, for a flat tube of unlimited breadth, the criterion 
pi205Un/e 18 ereater) than Dil 7) 2) 20 eee, emma 
37. This value must be less than that of the criterion for similar circumstances. 
How much less it is impossible to determine theoretically without effecting a general 
solution of the equations ; and, as far as [ am aware, no experiments have been made 
in a flat tube. Nor can the experimental value 1900, which I obtained for the round 
tube, be taken as indicative of the value for a flat tube, except that, both theoretically 
and practically, the critical value of U,, is found to vary inversely as the hydraulic 
mean depth, which would indicate that, as the hydraulic mean depth in a flat tube is 
double that for a round tube, the criterion would be half the value, in which case the 
limit found for K, would be about 0°61 K. This is sufficient to show that the 
absolute theoretical limit found is of the same order of magnitude as the experimental 
