280 MESSRS. PROUDMAN, DOODSON, AND KENNEDY ON THE DIFFRACTION OF 
which he found to be due to numerical errors on my part. From the larger number 
of values of 6 thus taken, graphs of Y 2 -\-Y 2 , Z l 2 + Z 2 2 (figs. 2, 4) could be constructed 
with confidence and no further work was necessary for kci — 1, 2. Tables X., 
XI. (*-« = 1, 2), and figs. 1 to 4, contain the results of this part of Mr. Doodson’s 
work. 
Sharing equally the labour, Messrs. Doodson and Kennedy then similarly recom¬ 
puted the case of kol — 10, and added the extra values of 6 which had been taken 
for kci = 1, 2. It then proved possible to construct rough graphs of Y u Y 2 , Z u Z 2 , but 
the uncertainties in the graphs of Y 2 + Y 2 2 , Z 2 + Z 2 were very great, owing to the 
magnification of errors involved in the squaring. In order to get more insight into 
the general nature of the results, they performed for kci = 9 all the work that they 
had done for kcc = 10, sharing it similarly. The construction of accurate graphs of 
Yi 2 +Y 2 2 , Z 2 + Z 2 proved as impossible as before, i.e., knowing only the points on the 
curves of figs. 25 to 28, indicated by small circles, the complete curves could not be 
drawn. 
Mr. Doodson then conceived the idea of calculating'the gradients of the curves at 
all the values of 0 taken, and formed the necessary series. Mr. Kennedy shared the 
computations with him, and the gradients were calculated at every known point of the 
curves for both <a — 9 and 10. These proved a great help. 
At this stage the results obtained for kci — 9, 10 were those indicated by the small 
circles and tangents in figs. 17 to 28. The small uncertainties in the graphs of Y ls Y 2 , 
Z u Z 2 , still led to considerable uncertainties in those of Y x 2 + Y 2 2 , Zy + Zd. 
Mr. Doodson then designed and carried out an examination of the way in which the 
results found differ from those given by the theoretical first approximation for large 
values of tea. The sections on analysis of results and interpolation give this work. 
Some of the methods proved their utility by leading to the detection of certain 
errors in the previous work, and all of them were used for interpolation purposes in 
constructing the final curves. 
Onfy the curves, as finally constructed from all considerations, are now given. 
In the work of drawing up the results, all the curves have been drawn by 
Mr. Doodson, while most of the tables have been written out by Mr. Kennedy. 
Only a selection, however, is printed ; the remainder are in the possession of the 
Royal Society. For instance, Tables T. to VIII. are only printed in so far as they 
refer to tea = 10, while Tables XII. to XIX., which refer to the gradients, are 
: omitted altogether. 
General Formula. 
2. For the theory of the problem we shall quote a paper by Dr. Bromwich on 
The Scattering of Plane Waves by Spheres.”* 
* This is a paper which Dr. Bromwich communicated to the Society at the same time as the present one. 
