502 Mr. 0. G. Darwin on Collision of 



right angles, and no particle can be deflected through more 



than a right angle. 



1 e 2 W 

 v = Q^Ntw v^ -^r™- 4 cot (j> cosec 3 <f>. 



But to this number we must add the helium atoms which 

 have been set in motion so as to strike the screen. They 

 number 



Since the energy of ionization is negligible they have velocity 

 Y cos 6. 



To find the whole number on the screen we replace 

 by <£ and get 



1 e 2 W 

 v = Q/Ntt ^i -^jy 4 cos <fi cosec 4 cj>(l -f tan 4 <f>) . 



The recoiling particles should be quite indistinguishable from 

 true a. particles. Indeed there is a reciprocal relation. I£ an 

 a. particle goes in a direction </> with velocity v while 

 simultaneously the nucleus goes in direction 6 with velocity u, 

 then it is possible for an a particle to go to 6 with velocity u 

 while the nucleus goes to <f> with velocity v. 



Case III. m< M. The a particles pass through hydrogen. 

 We have actually ??i = JM, and this special case is the only 

 one that need be studied. 



m? — M 2 sin 2 becomes negative i£ <£>14°29'. Thus no 

 a. particles are deflected through angles greater than this. 

 For less angles there are two types of a particles determined 

 by the two signs of the ambiguity. For example, in the 

 undeflected direction there are a particles which have not 

 been in collision at all, and also those which have struck a 

 nucleus perfectly straight and have then followed on. Thus, 

 in general, at any angle (/> there are a particles of two 

 velocities : 



v= Vl[4 cos <j> + v 7 ! - !t> sin 2 </>] 

 and 



V :==Vl[4cos (/>- y 1-16 sin 2 <£], 



and to each corresponds a different value of 6 which is given 



by the two signs in (3). 



Thus 



i 2i?2 ( cot $ + ^/cosec 2 ^ — 16) +(coc <£— v/cosec 2 <p — 16 



v— QNtco ^-. ^To cosec 3 cf> y— \-. — — 



^ V M 2 ' y/ (cosec 2 <£ — lb) 



