362 Prof. T. H. Havelock. Wave Resistance: Some 
variation of the resistance I with the velocity v, the Jollowing method is 
sufficiently accurate, at least for illustrating the main features. 
Take as a definite example, h = 2f; then, writing a for 2qf/c?, we require 
to calculate the value of a* [sect pe “4 cos? (a sec p) dp for various values 
of a. 
The integrand can be obtained without much trouble, and it was found 
sufficient to calculate its value at intervals of 10° throughout the range from 
0 to 7/2; the mean value was found from half the sum of the initial and 
final values together with the sw of the intermediate ones. In the course of 
these calculations, we have material for obtaining the value of 
7/2 : 
a’ | sec? pe 28°'b dh 
0 
by the same method ; but this integral is equal to 
fade {Ko (a/2)—(1+ 1/a) Ky (a/2)}, 
and we can find its value also from the tables of Ky and K, mentioned in § 4. 
By comparing results we obtain some idea of the accuracy of this method of 
numerical integration. The calculations can be lightened for present 
purposes by choosing, from general principles, values of « which correspond 
to important points on the graph. 
By this method we obtain values of R for different values of c, for this 
particular case. The result is shown in the full curve in fig. 2; the scale for 
R is arbitrary, the unit for cis the velocity \/ (gf). The dotted curve is a 
mean curve, and is equal to R:+ Re in the notation of this section; that is 
it represents the sum of the resistances due to the two systems, ignoring any 
interference effects. 
The graph is of interest in its exhibition of the typical humps and 
hollows, occurring in general when 2z7rc?/g is 2 sub-multiple of 2h. ‘The 
154 
