68o P.IKKI LAND, nn: NOKU '!:(. IAN ATKOKA roi..\uis i:xi'i IM IIO.N, 1902 '903. 



that is to sav when the particle is on a boundary-circle. Hence it follows that if the- particle at a certain 

 moment is retri ating fn in the globe, it will continue to do so until it comes to a boundary-circle; but 

 it will touch this and then turn inwards. 



It' we imagine a particle that is expelled from the magnetic equator of the globe, and assume that 

 after a limited time it comes to the nearest boundary-circle, it will move back to the globe again, aloni? 

 a path that is symmetrical to the one by which it moved out, i. e. the outward and inward going paths 

 lie symmetrically about the radius vector to the point on the boundary-circle in which the tangent 

 takes place. 



The correctness, of this is immediately seen when it is remembered that to a given value of r 



there are onlv 2 values of , which are equally great with opposite signs. 



tiff 



If therefore an ejected particle is not to return to the globe, it must move in such a manner as 

 never to reach the nearest boundary-circle. 'I bus it will move along a spiral with constantly increasing 

 distance from the globe, approaching the boundary-circle asymptotically. 



Let us now consider the integral 



'''/..!/ + ai 



= ( r 't.M+ar i/r 



'' '' 1 r,-4 . . o, it -3 J. 



- (a i- + IM )t 



where i' is the radius of the globe, and if A the value of if for r=r , and endeavour to find the 

 condition for the existence of such a spiral curve. If r= >', , indicates the smallest boundary-circle 

 (provided there are any Mii.li, i. e. that (1Y) has at least i positive root), then the Integral must be 

 infinite for r = >\ . 



Now it will immediately be seen that if r, is a single root in (IV), the function under the integral 

 sign may be written in the form 



1 /(r}, 

 \r r, 



where /(r) remains ordinary in tin: vicinity of r r t , so that we may put 



whereb the function under the integral sign assumes the form 



If we multiply by <lr and integrate indefinitely, it will at once be seen that the function of r 

 resulting from the integration will not be infinite for />',, and we therefore have no spiral curve ol 

 the kind required. 



It, on the other hand, r= ;, is a double root in equation (IY|, the function under the integral 

 sign will have a pole of the first order for r i\ , and then, as is known, the function will be logarith- 

 mically infinite for the same value. In this case, then, we obtain a curve of the required nature, and 

 also, as will be easily seen, if r = /, were a root of higher multiplicity, 



The problem is thus reduced to finding the condition for equation (IY) having a double root, 

 which is ^> r,, . 



It we confine ourselves to the consideration of particles that are expelled normally from the globe 

 in its magnetic equator, with an initial velocity r (l , we obtain 



A/A 



i> = Vc- -" and (' ir r] = " +^ = 



V ,// / " ,-- ' ,- 3 ' 



