First write G{6) in the form 



G{^) = (cos'0) [sin ea-e^sin'd)-''^']''. 



We wish to retain terms to e°, but no higher. Hence we expand the radical in (25) to 

 powers of e' since for n = 1, equation (25) will be multiplied by e^ as seen from (24) . Using 

 the binomial formula for the expansion we can write (25) as 



G(6I) = (cos^e) (sin d + Vte' sin^ 6 + ( V^) e" sin = + (V,e) e' sin' d)'\ 



To retain terms in e° we will need the first four terms of the expansion (24) and hence 

 three derivatives of (26). Now 9 = arc tan x,4^ = _L_ = cos^0, ^ ^ = - 2 sin ^ cos'0, 



dx 1 



d'0 

 d? 



2(3 sin'0- cos'0) cos^g. 



dG dG dd 

 dx dd dx 



d^G /d^G\ /d^Y /dG\ /di 

 dx' \dey \dx/ Wy Ids 



d'G /d'G 



d7 \de' 



d'G 



d^ 



d0 



dx 



d'G 



dd 



cos 0-21 1 sin 



\dd/ 



d'G 



d9- 



d9\ /d'9 

 dx/ \dx' 



d'9\ 



{ cos'0-6( 



\dd'/ \d9'/ 



dG\ 



cos sin f + 2 ( — 1 (3 sin'0 - cos' 

 d9/ 



(25) 



(26) 



(27) 



(28) 



(29) 



Because of the factor e' as a multiplier in (24), we can assume the following terms for (26) 

 for n = 1, 2, 3, 4: 

 n Gi9) 



1 (cos'0)(sin 9 + 'A e' sin' 9 + (3/8)e* sin' 9 + (5/16)e' sin' 9) 



2 (cos'0) (sin' 9+ e^ sin" (9 + e" sin' 9) 



3 (cos'0) (sin' 9 + (3/2)e' sm'9) 



4 (cos'0) (sin"6l) 



The terms of (24) are now formed by finding the derivatives of G{9) with respect to 9 using 



the appropriate form of G{9) from (30) and finding 



dG d'G d'G 



— , , by means of (27), (28), and (29). 



dx dx' dx' 



(30) 



17 



