496 Mr. C. Gr. Darwin on the Collisions of 



by quadrature and depend only on —. We denote them by 



Ht)> Mi)- 



Then J = P W _A 1 g +i A 2 g + ... 



=P(0 o _A 1 )+|(A 3 -A 1 2 )^ + ... . (3-7) 



The solution can then be completed by successive approxi- 

 mations. First, neglecting second differentials we obtain a 

 curve for P(#) from the experimental values of v. Then, 

 by means of the method of finite differences, we calculate 



d 2 F 



j£~2 and so get a correction term for v. In the actual 



CIUq 



curves analysed it was found unnecessary even at the point 

 of greatest curvature to make any correction for second 

 differentials. This implies that no discontinuities were 

 found. It may here be said that a slight modification of 

 our method would be capable of dealing with any of simple 

 type that might occur. 



In obtaining the P, 9 relation we neglected the loss of 

 velocity of the a-particles in their passage through the gas. 

 If this is to be allowed for, we have to use the accurate 

 equation 



y o Jo 

 taken between the limits given by (3*1) instead of (3*2). 



Assuming a Taylor expansion for P in Y, we can write 

 this as 



=!{*<'• y >-s; v fv*}*- 



The process of approximation could be applied to this, 

 though it would be very cumbrous. We should require a new, 

 more accurate, set of carves like that in fig. 3, but calculated 

 from (3*1) instead of (3*2). There would now be two para- 

 meters -7- and - , both influencing the calculations. We 



4r 



should then re-evaluate A T etc. We should also need from 



our previous solution to get relations between P and V, by 



taking together the experiments with a-particles of different 



*ip 



velocities. In this way ^^r could be found, and so the 

 second term evaluated, and applied as a correction to v. 



