A PLANE ELECTROMAGNETIC WAVE BY A PERFECTLY CONDUCTING SPHERE. 299 
In using the functions a u a 2 , &c., considerations of the possible forms for the 
theoretical higher approximations for large values of tea have been continually in 
mind. Take Y l5 for example ; the first approximation being Y x = | -kci cos 0, let us 
assume an expression of the form 
Y x = A ( 6 , kci ) cos {<£ (0, kci )}.(15) 
Then in the first approximation, we have 
A (0, KCl) = \kC<j, d? (0, kci ) = 0. 
Let us now suppose that higher approximations are got by taking 
A (6, kci) = \ko, + A 0 (6) + ——, $ {6, ko) = 0+ , . (16) 
\kCL ) 
kCI 
tea 
where the new functions of 0 are small compared with kcc. The omission of a term 
independent of kcc in $ (6, ko) is necessary if YJwkci cos 0 -> 1 as kci -> co. 
To the same order of approximation, (16) yields 
m = f« 0 + —) cos ©+ (b 0 + — ) sin 0,.(17) 
\ K(ll \ kOj} 
where a 0 , a lf b 0 , b 1 are functions of 6, being calculable in terms of those in (16). 
We then obtain, from (13), 
ci 0 + 
180 
19 + 
sin (38 cos -gfi) ] 
sin (2 cos ^6) j 
by sin (18 cos |-0) sin (20 cos |-0) 
90 sin (2 cos ^6) 
(18) 
and a similar expression for /3 V 
It is interesting to compare this hypothetical approximate expression for ay with 
what is known about the exact value of this function. Since the coefficients of c+ 
and by in (18) are rapidly oscillating functions of 6 with small amplitude, the 
suggestion immediately arises that the smooth curve a x of fig. 13 is the graph of « 0 ,' 
19 
or perhaps of cs 0 + -— ay, and that the ripples of the same figure are approximately 
180 
represented by the remainder of (18). 
From 6 — 150° to 9 = 180° simple graphical interpolation of Y 1? Y 2 , Z l5 Z 2 has been 
relied upon. 
Table XXYI. contains all the interpolated values of Yj, Y 2 , Z 1} Z 2 , Y 1 2 + Y 2 2 , Z x 2 + Z 2 2 
that have been used. 
