A PLANE ELECTROMAGNETIC WAVE BY A PERFECTLY CONDUCTING SPHERE. 293 
preceding tables, at the values of 0 taken. Tables XX. to XXIII .* give steps in the 
necessary computations, and tigs. 5 to 12 show the results plotted against 6 from 0° 
to 150°. These curves show that for this range the first approximation is very good 
for Z l5 Z 2 . In fact, the curves for £, were easily drawn and gave interpolated values 
of Z,, Z 3 to three places of decimals, the last figure being approximate. The functions 
i] 2 have a greater number of oscillations and greater amplitudes, and there was in 
places some little doubt as to the exact forms of the curves. By careful examination 
and comparison of corresponding curves for the two cases of tea = 9 and 10, this 
difficulty was overcome. Such a comparison was more fruitful when the curves were 
plotted with 0 as abscissae, and this was done with t] U t] 2 , Cj, C 2 , though the curves are 
not given below. Corresponding curves are very similar but are not in phase with 
each other. In the range 6 = 0° to 90°, it was found possible to obtain Y 1} Y 2 to at 
least two places of decimals, with very little doubt. 
It was the work just described which led to the detection of errors mentioned in 
§§ 1, 3, How this was possible becomes evident on an examination of the formulae 
and numbers. For example, an error in Yj or Z x generally leads to a larger relative 
error in dYjdO and dZjdO, to a much larger relative error in t 11 or and to a very 
much larger relative error in dtjJdO and dr^/dO. Thus, an error might pass unsuspected 
in the graph of Y 1 (for example) but render the drawing of the graph of >/, impossible. 
Irregularities could also be detected in the curves plotted with 0 as abscissae. By 
these means it was possible easily to detect errors in Y u Y 2 , Z 1? Z 2 , of the magni¬ 
tude O'Ol. The actual finding of the errors involved much patient revision 
of the summing of the series. Their existence had previously been entirely 
unsuspected. 
In this part of the work it was necessary to calculate a number of values of 
cos 0 and sin 0. The method adopted was to find 6 such that 0/x = r + 0]/l8O, 
where r is an integer and Q x = 0°, 10°, 20° ... 170°. This yielded about eighteen 
points on each undulation, and a selection from these was made according to 
circumstances. 
* Tables XXII., XXIII., and figs. 5 to 12 are not printed. 
Table XXII. gives rji, r) 2 , &c. (xa = 9, 10). 
„ XXIII. ,, 0»/i /d0, di]- 2 /dd, &c. (xa = 9 10). 
Fig. 5 gives the graph of r/j {<a = 9). 
6 
7 
8 
9 
10 
11 
12 
>> 
rji (xa = 10). 
1)2 (xa = 9). 
?/2 (xa = 10). 
Ci ( Ka = 9)- 
Ci ( Ka = 1°) 
C 2 { Ka = 9). 
£2 (K£l = 10). 
