( m ) 



e 6 r s 



4j'r(fz= \n—ir)d((= — - (30) 



— 1 



So llic e({tii|)()tcH(lal surfaces iive oöl<ite eillpso/j Is of iriujlutioii abont 

 tlie direction of motion. 



If V ^ c, we have to dislinguish, wlietbei- the point in ({iiestion 

 lies in region 1, 11 or UI. 



lil). In tliis case there are two positive roots t^ and two positive 

 roots T.^. We distinguish them as r/, t/', tJ, tJ' and easily see that 

 they arrange themselves according to their magnitude as follows: 



T ' T ' T " r " 



Indeed if we imagine a diagram in which for the abscissa cr the 

 curves // =z R — a, y = E -{- a and the straight line // = cr are 

 drawn, the latter intersects the former curves in four points, viz. 

 tirstly, beginning from 0, the curve y = R — a, then the curve 

 y z^ R -\- a, tiien the curve y = R -\- a for a second time, lastly the 

 curve y := R — (c for a second time. These four points belong to the 

 values r/, t.^', t^", t/', before mentioned. Moreover the diagram 

 sliow^s immediately, that the triangle (R, a, cr) is possible only for 

 those values of t, for wdiicli either t^' <^r <^t.^' or t./' <C r <^ t/'. 



On this account we obtain from (13) and (14): 



We introduce by (24) the variable u. 



It is to be noticed in expression (26), giving t by u, that the 

 denominator 1 — [T is negative as well as the term a^i-\-iix\\\ the 

 numei'ator, the latter being so because we are in region 111. From this 

 follows, that the negative sign of the root in (26) belongs to the 

 interval of the larger values of t (t.," < t < t/'), the positive sign 

 to that of the smaller ones (t/ <^t <^ t,'). y\ccordingly Ave have to 

 put in the second integral of equation (31), which is extended 

 between the two largest roots t^' anil t/': 



c dr du 



a R \/[x + an [3)' + (1 — 1^) {f + z') ' 



whereas we have to use in the lirst integral Ihe formei- e(piati(»n 

 (28). It follows that 



