GEOMETRIC INVESTIGATIONS ON DYNAMICS. 305 



By combinations of these we get four additional relations 



dt\ dF, ,, dFi dFo dF2 dFz „ 



Pi- Pi^^ 0; />>- P^-^ = 0; V^- 2^2 T—= 0; 



dpz dp3 dpi dpi dp-i dpi 



dF^ dF, „ 



djh dp2 



These 12 relations may be written in the equivalent forms 



d 



dp 



(p-iFi-p.F^ =0; ^ (Mi - PiF^ ^0; /; (P3i^2 - P2Fz^=0; 

 - Uf,- p,Fs^ = 0; ^ CpiFs - iJ^F^ = 0; ^ (luFs - pzF^ = 0; 



^ (piF,- p,F^ =0; ^ Uf, - p,Fi\ =0; ^ (^7.3^1 - pii^s^^O; 



^ (^3^:- /^iFs") =0; ^ (i^4F2 - p,F^ = f^; ^ (M2- p2/^4)=o: 



Hence 



7^2^! — plFo = OCn; P3F2 — p-lFz = a23; ^^4^3 — ^^3^4 = «34; 



^ piFi — piFi = 0:41; P3F1 — piF-i = a-iz; PiFi — p^Fa = 0:24; 

 where, e. g., 



(47) ai2 = aviipu Pi, Xu x^, .T3, .T4); a-iz = a'2z{p-2, Pz, .Ti, 0-2, .T3, .T4); etc. 

 The vanishing of the last three brackets in (45) gives 



SFi px dFi dFo Pi dFx dFs ?J3 dFo _ dFi pi dFz 



(48) 



dpi pidpi dp2 pidpo dpz P'ldpz dpi pzdpi 



These are conditions on the forms of the a's appearing in (4(3). To find 

 these conditions we may proceed as follows: in 



dFi pi dFi _ dFz _ pzdFi 

 dpi pidpi dpz p-idpz 



dFi , PidF^ , , 

 we may replace bv , and get 



dpi " Pi dpi 



dFi dF.2 dFz dFo 



Pi- Pl^ = P2- 2^3- 



dpi dpi dpz dpz 



