a-Particles with Hydrogen Nuclei. 



495 



If the area observed is A sq. cms., then the number seen 

 will be 



v = v \ l 'Pd i v, (3-4) 



r here 



Aqnz.. 



iirr- 



(3-5) 



In this equation v is a known function of -. ---, P is an 

 unknown function of 0, while x is a known function of y 

 depending on the parameter — . This is an integral equation 



of the ' first kind ' of the type known as Volterra's, with 



dx 

 'kernel' -j^ m The present writer is not competent to 



discuss the general theory of such equations, but it appears 

 that the following process gives quite a satisfactory solution. 

 This is because the limits of integration O and l are in all 

 cases not very far apart. 



Fig. 3 illustrates an x, 6 curve, with the difference between 



0-- * x 



O and 1 somewhat exaggerated. The tangent at O is- 

 always horizontal. Assume that P is a function of 0, that 

 can be expanded in a Taylor series. Then in the field of 

 integration 



and so 



v 



%/ e 1 dVoJe, ^ «#o Je 1 



The first integral is equal to unity, the second is the area 

 of ABC taken negatively, and the third is twice the moment 

 of this area about AB. The last two quantities are calculable 



