160 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
If wu’, v' ave indefinitely small the last term, which is of the fourth degree, may be 
neglected. 
Substituting in the discriminating equation (43) this may be put in the form 
apa |)( 222) oa. (Bm Vl] are Sea ea 
2pbUn _ 720 (: {ul ed ai (a) a Gi | * ee a ( i) ja (65) 
ie 3 [dy [ 3 {nt( = 2 — a, aT fay 
—ly 4 —2o 



Limits to the Periods. 
36. As functions of y the variations of @,, 8, are subject to the restrictions imposed 
by the boundary conditions, and in consequence their periodic distances are subject 
to superior limits determined by 20,, the distance between the fixed surfaces. 
In direction x, however, there is no such direct connection between the value of b, 
and the limits to the periodic distance, as expressed by 27/nl. Such limits necessarily 
exist, and are related to the limits of «, and 8, in consequence of the kinemetical 
conditions necessary to satisfy the equations of motion for steady mean-mean- 
motion ; these relations, however, cannot be exactly determined without obtaining a 
general solution of the equations. 
But from the form of the discriminating equation (48) it appears that no such exact 
determination is necessary in order to prove the inferior limit to the criterion. 
The boundaries impose the same limits on «@,, 8, whatever may be the value of il ; 
so that if the values of «,, 8, be determined so that the value of 
26bUn 
is a minimum 
for every value of nl, the value of 7/, which renders this minimum a minimum- 
minimum may then be determined, and so a limit found to which the value of the 
complete expression approaches, as the series in both numerator and denominator 
become more convergent for values of nl differmg in both directions trom 77. 
Putting J, a, 8 for r/, «,, B, respectively, and putting for the limiting value to be 
found for the criterion 


20b)Um di 
ke ihn | ae? (66) 
bo op! ( (da\? /dB\? | da\? PB 
: Me +B) +20) (F) + (5) |-+ (2) (5) bey 
SK, a i | ae ay) dy | dy ) i i (67) 
ar a (e@ — a) ay 
ye a Ney “dy ; 
when a and B are such functions of y that K, is a minimum whatever the value of J, 
and / is so determined as to render K, a minimum-minimum. 
