116 Proceedings of the Poyal Society of Edinburgh. [Sess. 
appropriate to Case II or Case III, whenever the ratio sjs 2 passes from an 
irrational to a rational value. The advantage of solving by means of the 
adelphic integral is that the forms of the adelphic integral corresponding to 
the three cases can be readily determined : and thus the difficulty is 
removed before the adelphic integral is used in order to obtain the complete 
expressions for the co-ordinates in terms of the time. 
(Issued separately April 30 , 1917 .) 
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