﻿226 Mr. L. Vegard on tJte Properties of 



great number of points of discontinuity corresponding to 

 orbits which do not pass on towards infinity, and the curves 

 fiirf) nll< ^ f-2 (7) break up into a number of separate parts. 

 These curves have not yet been exactly determined ; but 

 from Stormer's investigations we see this much, that in the 

 interval (I 2 j ^r will oscillate inside a fairly small interval, 

 while the angle <£>, especially for orbits near those returning 

 to the origin, may assume enormous values. In other words, 

 the orbits of this interval may turn several times round the 

 magnetic axis before they strike the earth. 



As such a large variety of orbits corresponds to a very 

 small interval of 7. very small variations in the initial con- 

 ditions may cause great changes in the form and position of 

 the aurora?. Consequently, we sh ouldexpect that aurora?, as 

 far as they are produced by rays following orbits of the 

 interval (I 2 ), would on an average be spread over all hours 

 of the day, and we should not expect aurora? formed in this 

 way to produce very distinct maxima or minima. At the 

 present stage of mathematical investigation we are not able 

 either to prove the existence or to determine the position of 

 such possible maxima corresponding to the interval (T 2 ). 

 For the aurora? possibly belonging to the interval (li), 

 however, we can estimate the position of the diurnal 

 maxima. 



A number of orbits of the interval (I{) are given in fig. 6, 

 which is a photograph of a curve model made by Stormer. 

 The orbits are all supposed to come from points situated in 

 the same plane through the magnetic axis, and at a distance 

 from the origin equal to that between earth and sun. Curve A, 

 fig. 7, shows the relation between <& and yfr in this interval. 

 When 7 passes from to —0'93 <I> increases from 0° to 306°. 

 The position of the spot of precipitation is determined from 

 theory by the position of the sun relative to the magnetic 

 axis, and will be given by the points of intersection between 

 the curve (A), fig. 7, and the straight line (B) given by the 

 equation \|r= +S ni , where 90° — 8 m is the angle between the 

 direction to the sun and the north direction of the magnetic 

 axis. 



The distribution previously found is derived from obser- 

 vations during the winter. In that case B m is negative, and 

 the line B will intersect the curve A in four points, corre- 

 sponding to four diurnal maxima. The first and the last one 

 occur in the day and escape our observations. The points 

 viy and m 2 are about symmetrically situated with respect to 

 magnetic midnight and correspond to night maxima. 



Thus the observations are so far in agreement with theory 



