ÖS'VERSIGT AF K. VETENSK.-AKAÜ. FÖRHANDLINGAR 1 900, N:0 U. 1 075 



singular fact tliat to eacli value of C tliere exists at niost a 

 Single curve of each faiuily. 



To demonstrate this I recall the integral of Jacobi (4). 

 This is 



<30) ^"- + V'=-2ß— C. 



It is easy to express C trough the initial values ^„ and r^^. 

 We have 



§ = 2aj cos v^t + 2a.y sin V.-J, 

 -»]= 2/?, cos V.J + 2/?o sin vj . 

 Hence 



§' = — 2v^a^ sin v^t + 2v^a., cos v4 

 rj' = — 2j'o/^i sin v^t + 2,v^ß.^ cos v^t . 



If we put in these equations t = O, we get 



(31) ' 



^0 = 2/?. 



and with the aid of (23) we can now express ^'^ and 'j^'^ through 

 «j and ß^ and hence also through ^^ and '^^^. We have 



^ - da 00 



I d-P\ 



It is thence 



d'^n 



2'"^'« = ^6^'° + (^' + "#)^o 



2 ^'•-iiJl 



se equat 



4^'K + O 



dadh'^''' 



And from these equations we get 



öadbj \ da- 



d^n \2 



dadb 





I: 



(^2ß 



^2^3" 



''" + TT^ + ^' + ;37 2 



'o 

 2 .^ 



dadb 



So^o 



