696 HIKKKLANI). T11K N( iRWKCU AN AURORA POLARIS KXPKDIT1ON, igO2 1903. 



and 



(d'l /"(>-,) = >- ( ,z' sin -j- //<;', . 



The effect of the equation (c'( is that 



^ ," < *'f,>'o < -," - 

 it r, is to have a positive value ]> r u . 



(d'| will certain!}- he satisfied it' 



V ." ''i < " '' ('',) < W + \ ," '-, = ' ; ' - 



Moreover r, must he the smallest positive value of r that causes the expression under the square 

 root sign in the y>-integral to vanish. 



The and derivative of the raclicand has for r = r the value 



As however 



'd I. 



., , "C *' > an d '',,''0 sm c 'o ~~ ^'"( ; 'i) <C ^ > 



/*/ ^ "/ > f 



the second derivative is negative. The ist derivative then becomes positive for i'<^>\, and conse- 

 quently the radicand itself negative for values of r <^ r t . But then the particle cannot approach any 

 boundary-circle. 



On the other hand, the orbit can certainly become a conic section at last. 



Let us consider the simple case in which the particle retains its charge until it comes very near 

 the boundary-circle, assuming that it tends towards one, but then suddenly loses its charge. 



The changes from in-going to out-going motion, or vice versa, will then take place for those values 

 of ; that satisfy the equation 



-)- V' j i" - - 2 ft r r\ v'\ siir ft, , 



where r\ is very nearly the radius, r,,, of the boundary-circle, ;', very nearlv the velocity, v (] , in the 

 boundary-circle, and , very nearly - ~. For the sake of simplicity we may put 



!C 



~ ' a < ' ' i == ''v> a i ~ 2 ' 



whereby we obtain 



The discriminant of this equation is 



^>'u 

 and consequently the roots in the equation are always real. 



