26 Art. 3. — K. Hiiayama : 



The parts, or tlie p«irt as tlie case maybe, of A inside of Bo 

 and outside of ^i are tlie real patlis of the point (x, e). In Fig. 1 two 

 parts, one on the negative side of x and the other on the positive 

 side, are separate. The argument 6 may take any value in each 

 part. 



But when £ becomes larger or C becomes smaller, these two 

 parts approach each other until A touches Bi, in which case they 

 are connected. After this A does not intersect B^, as in Fig. 2, 

 and the argument 6 never takes the value n or—-. It increases 

 from to the maxinmm value do and decreases, passing through 

 again, until it reaches the minimum value — ^o- Then it increases 

 again and so forth. This is a kind of libration.^^ 



20. We can distinguish six types of the 7noiio7i, namely: — 



1 Revolution on the negative side (of .t), 



2 ,, extending on both sides, 



3 ,, on the positive side, 



1 Libration on the negative side, 



2 ,, extending on both sides, 



3 ,, on the positive side. 



21. In order to determine the limits of C and ^ for each 

 type, we need to consider some singular cases. 



1. Condition of the contact of A and ^o or ^i. Difïerentiat- 

 ing the equations (16) and (18) with respect to a- and equating 



-T-we get the condition 

 (19) ex=±^ 



The same equation results from (17) putting cos d=± 1 and -rr =0 

 in it. To find the relation between (J and E, eliminating x and 

 e from the equations (IG), (1<S) and (1Î)), we get 



1) Libration is clofinecl as a case of motion in which the value of the argument is limited. 

 This definition is slightly different from that of Prof. Brown (Month. Not. Ixxii p. 618). 

 The term rej'o/Hff on is used after M. Callnndreau (Annal.ilelOlis.de Paris, IVlémoires Tome 

 XXII). 



