64 Mr. G. F. C. Searle on the Impulsive 



Now cosd 2 =(v—h 2 )/u 2) and hence 



fcos 2 dco 1 f Cdxo Cdw 1 

 J hjiz ~ u 2 1 J /ij/ig J h x J * 



But, by (7), v J^ = iFK W2 ,«) 



Cdo _ 9 f» sinfl^flx _ 2tt 

 J ^i ~" Jo *>— w i cos #i ~~ u i l ' 



and 



where 



T , v + u x , ,. , V+W 2 



1^ = log- and L 2 = log — 



u— W.i 



V — ll-L 



With the notation of § 10, 



cos # 2 = cos 0i cos a + sm $i sm a cos $> 

 and thus 

 Tcos S 2 dw _ rr(cos 6i cos ol -f- sin 1 sin a cos (f>) sin d^O^cf) 



where <£ goes from to 2w and 6 ± from to it. The term 

 involving cos</> vanishes on integration with respect to <£. 

 Writing (v— 7*i)/i/i for cos 1? we obtain 



fcos fl 2 ^6> _ 2tt cos a f, f" sin fl^ f ff sin S 1 dO l 1 



a \ 2v __ 

 | v 2 — U-? 



;7rcos 





Writing (v — li 2 )/u 2 for cos# 2 , we obtain 



_ 2ir [ £ 7r s'm6 2 d6 2 f * sin 2 ^2 "I 



J cos 0$ 



«2 



n 



Ao 2 



Jo ' *. J 



= 2tt J 



2w 



w 2 |^—w 2 2 w 2 J 



Collecting these results, and substituting for F(n 1? n 2i r a) 

 the value derived from (8), we have 



P 2 + P!COSa= — + — 



f (y 2 — Ui) COS a __ V 2 — w^ COS a 1 p 

 I 2wi 2 u x u 2 J 2 



. y 2 — ?i 9 2 T . t"(?/i— i/ 2 cosa)~| 



Jw 2 - MiW 2 J 



