522 T. H. Havelock. 
For a given value of ¢’, the value of (19) was found for about a dozen values 
of «, so that the graph could be obtained with sufficient accuracy for our 
Iie. 2. 
purpose over a range of «, that is of «gr, extending from 0 to 18. In each case 
the value of (19) was obtained by evaluating the integrand at intervals of 0-1 
for ¢ for a sufficient range of ¢ until, by reason of the exponential factor, the 
remaining terms became negligible. Sets of calculations were made for two 
values of @, that is of kof, namely, 4:0 and 0-5; in the latter case it was 
necessary to take 40 or more values of the integrand in each case, but a smaller 
number sufficed in the former case. The value of the integral was obtamed 
finally by using Simpson’s rule. The collected results are shown in the graphs 
of figs. 3 and 4, the curves being lettered in agreement with the radial lines 
of fig. 2. 
Fia. 3. 
Fig. 3 is for «f= 4, or c= 4)/(qf). Consider first the radial lines within 
the limits of the ideal wave pattern, namely, A, B, C. In this case, though 
B and C were calculated separately, there was not sufficient difference to show 
on the graph without confusion and so B has been omitted ; A is the central 
line and C, at an angle of 19° 26’, would be the cusp line of the simple theory. 
We may picture the waves in the present case as chiefly transverse waves, 
295 
