u Particles ivith Light Atoms. 501 



Let N be the number of atoms per c.c. Let Q a particles 

 pass in each second through a foil of thickness t. Then 

 Qt N 2irp dp of them pass within a distance between 

 p and p + dp from some nucleus; and all these undergo a 

 deflexion (/>. Suppose that v are observed per second within 

 a solid angle &> at inclination <£. Then 



v = QiNeo . P , J j, • 

 sin <p a<p 



If we substitute the value of p in terms of $ deduced 

 from (4) and (3) we get the accurate formula for v. viz. : — 



1 e 2 W 



[cot <f> ± \J cosec 2 <£ - ( — J 

 Q^^^cosec^ -^ ■ 



.(6) 



3. We must now remove the ambiguity of sign. This 

 depends on the values of M and m. 



Case I. m>M. This will refer to all substances except 

 hydrogen and helium. Consideration of the special cases 

 <f) = and (f> = 7r shows that the upper sign is to be taken. 



If the expression is expanded in powers of — we find 



m 



^ XT le 2 E 2 lf 4 <£ /MV / 3 i a A/MV \ 



The first term of the expansion is that given by Rutherford. 



Since cosec 4 ■j- is always greater than 1, the result is correct 



M 



to 1 per cent, with these three terms, even when — is as 

 1 m 



high as J, the value it would have for carbon. From (2) we 

 see that the velocity of the carbon nucleus may rise as high 

 as \Y. Such a particle might possibly be perceptible, but 

 it is doubtful whether the shock would free the nucleus of 

 all its electrons, and the experimental conditions to reveal 

 it are rather hard to imagine. 



Case II. m = M. The a particle now travels through 

 helium. We have still to take the positive sign in the 

 ambiguities. Then we get : — 



77" 



v = V cos (/>, u = V cos #, ({>= -5 — 0. 

 The a particle and nucleus diverge from one another at 



