A PLANE ELECTROMAGNETIC WAVE BY A PERFECTLY CONDUCTING SPHERE. 295 
5. There remains the discussion of Y 1} Y 2 for the range 6 = 90° to 180°, and of 
Z l5 Z 2 for the range 6 = 120 ° to 180°. For portions of these ranges a method was 
used which consists in decomposing t lu > ?2 , £ 2 into functions for which graphical 
interpolation is easier. 
For both kci = 9 and 10 , take 
>?! = a 1 cos 0 + /3 : sin 0, £ = y x cos 0 + ^x sin 0, 
rj 2 = a 3 cos 0 + ft 2 sin 0 , = y 2 cos 0 + S 2 sin 0 , 
( 12 ) 
where a 15 a 2 , /3 1 , fi 2) y l , y 2 , S 2 are functions of 0 only. 
These functions, thus defined, are perfectly determinate, but they can only be used 
for net = 9, 10 . Together with their derivatives, they are calculable for the tabulated 
values of 0 . We have, in fact, 
a, = 
a 9 
A (;/]/sin 0) 
■ A (cot 0) 
A (%/cos 0) 
A (tan 0) ' 
■ / 3 3 = 
A (>;i/cos 0) 
A (tan 0) 
A (^/sin 0) 
A (cot 0) ’ 
(13) 
with similar formulae for y lf y 2 , S u $ 2 , where AF (kci) denotes F ( 10 ) —F (9). Also, 
da x 
A 
ffa/sin e) 
+ -4- 
do 
180 
Mi = 
A 
f §21 /cos a 
7T 
VS6> / 
180 
sin {aj + ^A (kcc cot 0 )} 
sin 6 {/3 1 + ajA (kci tan 0)’ 
A (cot 0), 1 
A (tan 0), 
(14) 
with similar formulae for the remaining cases. 
The values of the functions themselves are given in Table XXIY. for the range 
6 = 0 ° to 150°, and are plotted by means of small circles in figs. 13 to 16. As will 
be seen, it is possible to draw smooth curves with few undulations approximately 
through these points. The complete graphs of a,, a,, &c., differ from the curves of 
figs. 13 to 16 by a number of ripples. An attempt has only been made to draw these 
ripples in those parts of the curves actually required for interpolation. For this the 
gradients were necessary, and the values of the derivatives used are given in 
Table XXY. With the limited data of Tables XXIY., XXA 7 . the drawing of the 
ripples is not perfectly determinate, but it was assumed that the graphs of rj u >/ 2 , &c. 
(and of Yj, Y 2 , &c.), are without ripples, and after a number of trials the ripples of 
the graphs of a 1} a 2 , &c., were drawn in such a way as to secure this, but they are not 
given in the figures. 
YOL. CCXVII.-A. 2 T 
