60 Mr. G. F. 0. Searle on the Impulsive 



Solving these equations, we have 



-p X 2 — XiCOSa p _X X — X 2 cosa 

 snra sura 



The magnitude of P is given by 



= P 2 X 1 + P 1 X 2 



^ X 1 2 + X 2 2 -2X 1 X 2 cosa 

 sin 2 ol 



If Jwe substitute for M x , M 2 , W^ and W 2 the values given 

 in (37), (38), (32), and (33), and write 



, t'-fl^ T , V + U 2 



log— - i =L 1 , log 



to v — u x ^v — u 2 



and make a slight re-arrangement of the factor of L 1? we 

 find that 



_, -~ W Ua r f (v 2 — W a 2 ) COS a V 2 — UiU 2 COS OL \ T 



2u 2 * u x u 2 J v ' 



Similarly, we find 



-d , -d W , U r f(v 2 — w 2 2 )cosa v 2 — Mucosa") T 

 P! + P 2 cosa= + — \ ± ^ — — yL> 2 



Ul V L L JW UiU 2 J 



+ ^ Ll + <"*-"' cosa > l. . (43a) 



We can, without difficulty, write out the value of P 2 , but 

 the expression becomes very complicated in the general case. 

 We shall therefore consider its value in special cases only. 



The component parallel to Uj of the momentum in the 

 pulse is Pj + Pgcosa. If we make u 2 =0, we obtain the 

 momentum in the pulse which is formed when u x , the initial 

 velocity of the sphere, is suddenly destroyed. The expression 

 for P]+P 2 cosa (not P^PxCosa) becomes indeterminate 

 when we put w 2 =0, but this difficulty can be overcome by 

 first expanding L 2 in the form 2w 2 + 2w 2 3 /3 + . . . , and then 

 putting w 2 = in the terms which do not cancel. Using the 

 value of W x given by (34) and using m 1 = ?? 1 v_, we find for 



