u-P articles with Hydrogen Nuclei. 491 



relation between P and #, or, allowing tor experiments with 

 different velocities of the a-particles, between P, 0, and Y. 

 This is the collision relation. 



Our first task then is to find the collision relation for 

 Rutherford's experiments. Afterwards we shall have to 

 calculate the projection diagrams for our various models and 

 deduce from them the corresponding theoretical collision 

 relations. It will be well, therefore, to consider some of the 

 characteristics that this relation may be expected to have. 

 Let us suppose that the forces between the nuclei do not 

 include any which can produce impulsive changes of velocity, 

 so that the distribution of the curves in the diagram is con- 

 tinuous in the widest sense of the word. In spite of this 

 continuity P may have discontinuities of its derivative. For 

 example, in fig. 1 the P-curve would have a discontinuity of 

 tangent at 50°. For consider how P behaves as# diminishes 

 through 50°. For angles just above, it is shrinking at a 

 rate given bv the rate of decrease of the outer curves, but on 

 passing 50° a fresh area must be subtracted, that of the 

 small patch round Q. This leads to a discontinuity of the 

 tangent. 



in the case where both a and H were regarded as point 

 charges *, a relation was obtained from the orbits of the 

 form 



p = fi tan 6, ...... (2*4} 



where 



Here p is the length of the perpendicular from H on to the 

 initial line of motion of u. By this relation p can be 

 uniquely determined from #, so that the projection diagram 

 consists of a series of circles, and unlike fig. 1 has no 

 maxima or minima. We should therefore have P = 7rp 2 and 

 (2*4) would express the collision relation as it stands. Now 

 it is more convenient to deal with lengths than with areas, 

 so in the general case we shall define a length p by the 

 equation P = 7rp 2 , and shall use p instead of P. It should 

 be remarked that even in cnses where the projection diagram 

 is entirely composed of concentric circles, so that 6 can be 

 determined by p alone, it does not follow that p is the same 

 as p. For several values of p may lead 1o the same 0, 

 whereas p from its definition in terms of P is necessarily a 

 single-valued function of 6. The deduction of p from p is 



* Rutherford, loc. cit. The relation is proved with a slightly different 

 notation in C. G. Darwin, Phil. Mag. xxvii. p. 499 (1914). 



