438 Prof. T. H. Havelock. 
Denoting by Wo, Wi, 2, ws the solutions beginning with 1, «, «3, 23 
respectively, it follows from (12) that the boundary conditions lead to 
ri dwr3/da—rWv3dyn/de =0 (14) 
where « has to be replaced by p. 
Calculating the coefficients far enough to give sufficient accuracy for our 
purpose, we have 
Yn = a+ 23/3 !4 (343k) 09/5!4+(44 20K) a /7! 
+(5 + 70k + 33k?) a®/9!+ (6 4+ 180k + 366K) «4/11! 
+(7+ 385k + 2029K + 627k) a!9/13!+ (8 + 728k + 7832k? + 9672k3) 219/15! 
+ (9+ 1260k + 24030K? + 73500 + 16929k4) aM /17!4..., 
abs = 08/3! +4 20°/5!4(34 7k) a7 /7!+(44 36k) 29/9! 
+ (5+ 110k + 105%?) a! /11!+(6 + 260k + 8944?) a!8/13! 
+(7+525k + 4213k? + 2415k3) a!9/15 14+ (8 + 952k +1455 2k? 
+ 28968) al /17!+... 
Forming equation (14) we have 
2/3!+ 8p? /5!4+32p*/7 14+ 1288/9 !4+ (512+ 192k?) 8/11! 
+ (2048 + 22444?) 99/13 !4+ (8192+ 194562") p?/15! 
+ (32768 + 139264k?) p#/17!+ (131072 + 901120 
+129024k*) p'*®/19!+...=0. (15) 
Only even powers of & appear in this equation, thus giving a check upon 
the arithmetic ; further, the terms independent of & may be summed. Taking 
the least root of (15) as an equation for %?, we have approximately 
R= sinh 2p — 2p - 6) 
ee ee ) 
Itt Te 15! 17! at atk 
Instead of forming an equation for the minimum value of R, it is simpler 
to find it by trial. We find, with sufficient accuracy, that it occurs near 
p? = 11, and then, approximately, 
R = 96. (17) 
The corresponding value, found by Orr, for flow under similar conditions 
but with a fixed plane at the upper surface, is 117. We conclude then that 
flow in an open canal has a lower degree of stability than flow between fixed 
planes. 
Turning to experimental results, the number usually quoted for flow 
through a tube is 2000 approximately. This was obtained chiefly from 
