Electric Waves round a Large Sphere. 



531 



derivate of an exponent, 



T^-lX 1 * 



"Pn-Qnr-mO+i* 



+ e l 



, — <p nr -\-'2X 



™e+i*-) (44) 



where in terms of x — m/z, 



u = ix{2zx sin 0R n B. nr )i/a7r^, • • • (45) 

 and where the exponents have zero derivate when 

 x = sin 0/(1 — 2c cos d-\-c 2 )%, c = a/r. 



But when 6 is obtuse, sin -1 or and sin -1 x cannot have a 

 difference equal to 6. In fact at x = l, the solution ceases, 

 so that on the left of the plane rcos# = c, there is no point 

 leading to a vanishing derivate of an exponent. The value 

 of 7/0 would then, except in the region of transition as 

 previously indicated, be deduced from the integrated sum- 

 mation formula. It would consist of limit terms, of which 

 those at the upper limit would vanish in accordance with 

 previous theory. Since therefore the sum is to be deduced 

 from integrated terms at a lower limit, it is obvious that the 

 use of the asymptotic expansion of the zonal harmonic may 

 not be legitimate. The expression for yp given in (39) must 

 therefore be evaluated instead for points beyond the left of 

 the region of transition. But as written, the calculation 

 would present difficulty, as <f> may in its range take such a 

 value as to cause a zero derivate of the exponent. This may 

 be avoided by the use of the alternative expansion 



PM _2 C* sin mft d<j> 



r * W ~ TrJe v'Vlcos 0-cos $)' ' ' ^ b) 



If <£ nr — </>„ cannot be so great as ±0, a fortiori it cannot 

 attain the value $, which ranges from 6 to it. 

 Denning in this case an operation Gr(#) by 



2i sin 2 6 d_£* a>d<f> 



dd 

 then 



G(0> = 



ka-ir da J g <y 2 s/ (cos 6 — cos <f>) ' 



(47) 



yp = G(0)2" m(R n -R nr )^e l *>-*»r + e l 

 = G(0)2* u(e lV i+e lv >-e lV *-e lV *), 



n ) sin m<f> 



here 



u=^r (R,3.»)2, 



V 2 = 4>n — <l>nr + ™# + 2%,,, 



(48) 



2 M 2 



