

a-Particl.es with Hydrogen Nuclei. 505 



cases it is necessary to change p into p. Let p u p^ P% be 

 the three ordinates of p for any value of 6. Then 



p 2 =p l 2 -p 2 2 +p 3 \ 



as may be seen by making a sketch of the projection 

 diagram, Fig. 7 shows the collision relations for the three 

 values of V determined by ^/6 = 0'24, 02, O'l. The dis- 

 continuities are partly due to those of p and partly to the 

 conversion of p into p. In some ways the curves are better 

 than those of fig. 6, but they are still very unlike the 

 experimental. One part of the experimental curve A is 

 however well illustrated and explained. The existence of 

 its flat part implies a discontinuity of 6 as p varies. 



9. The Bipole *. 



Of the problems of orbits that are integrable, there is only 

 one of a more complicated type than those we have con- 

 sidered. This is the celebrated problem of Jacobi of two 

 centres of force ; so in order to push our investigation 

 further we are compelled to call in its help. Now we want 

 a body that acts something like a plate, and the field round 

 two equal centres of force is of course quite different from 

 this, but by considering one special set of orbits, we can deal 

 with cases of motion through a set of eqnipotential surfaces 

 not very unlike those of a disk* The comparison will be 

 carried out in the next section. Here we shall suppose that 

 « is split into halves at distance 2a apart, and that H is 

 projected towards them along some line at right angles to the 

 line joining these halves. This will give us a relation 

 between p and 0. We then imagine this relation to be true 

 for all azimuths and not merely in one plane. With this 

 assumption we convert p into p, and so obtain a set of curves 

 for the collision relation. 



The solution is found by means of the transformation 



x = a cosh f cos?;) fQ«"n 



y = a sinh f sin r\) 



and following known processes the orbits due to projection 



* This term is used in the sense of two equal charges, not as is usual 

 of two equal and opposite charges. 



