FLUIDS AND THE DETERMINATION OF THE CRITERION. 163 
value ; so that the latter verifies the theory, which, in its turn, affords an explanation 
of the observed facts. 
The State of Steady Mean-motion above the Critical Value. 
38. In order to arrive at the limit for the criterion it has been necessary to consider 
the smallest values of w’, v’, w’, and the terms in the discriminating equation of the 
fourth degree have been neglected. This, however, is only necessary for the limit, 
and, preserving these higher terms, the discriminating equation affords an expression 
for the resistance in the case of steady mean-mean-motion. 
The complete value of the function of transformation as given in equation (64) is 


SU eG dp, det, OF ee aBy, eR 
— 3 — —— 2 — 7a 
—-> lp The [dy [inl (1, TF [5= - )e dy aR ie al ie) (4.5 7 B al dy) 5 (71a) 
Whence putting U + wu — U, for w in the left member of equation (77), and inte- 
grating by parts, remembering the conditions, this member becomes 
Fe | ay (pu dy + © | we? Wy 6 3 5 2 0 > (78), 
in which the first term corresponds with the first term in the right member of 
equation (64), which was all that was retained for the criterion, and the second term 
corresponds with the second term in equation (64), which was neglected. 
Since by equation (35) 



3U 5 1 dp 
a MER lag eee 
we have, substituting in the discriminating equation (43), either 
1/dt (pH’) d; 2 (ie0 
Ae (pH’) Lae a f (wv)? dy) 
ee) p bo? dp __ 2b? [is Ke +o eh (79) 
3 2 pera we) bo Y ? 
ee ee * — | dy (we dy 
—bo —bo 
or 
du dp 
Lp ae Oe Meee ai) eo alc cs cud (OO))E 
Therefore, as long as 
