54 C. V. L. Charlier 



aud 



[u^] — I cos '■^^ sin ^d- + î/g) (— 3 -|- 

 (1 14) [ ?^'] = I CCS ^d- sin ( f j + (— 3 w^), 



[w^] = I cos 2^ sin ('^r, + «•,,) (- 3^^;' + co^), 



where in the right raember the relative velocities (m = — etc.) are introduced. 



[vwj. 



We have 



0 



In computing v^'w^' we make use of the formulae (18), so that 



(OH . . VW \\ 



v'w. ' 



\ i\ sin + cos — cos sin x)- ( cos <h — 



X {M'^sin^'S- + lOgCOS^ö' — cos sin d ( — sm '\ \/ ii^ -\- v^)] . 



Observing that 



27: 



l[sm^-H.^ = i, 

 0 



we get 



•27: 



~ J^^i'w^' = [v^ sin ^t)- + cos ^0-) (w^ sin ^t)- + ^f, cos -v)-) — cos sin '^d 



VW 



2% 



0 



and in like manner 



2n: 



1 r VW 

 2^ ^^'««^o' = {i\ cos ^')• + ^2 sin ^')■) cos + iv^ sin ^t)-) — cos ^â- sin ^0- — 



0 



Using these formulae we get 



[vtv] = v^ii\ (sin + cos ''ö-) + v.yW.^ (sin + cos *&) 

 + 2 sin cos (^1^2 ^2*^1) — cos ^'^^ sin vio 

 — v^iv^ — V.jW.^ , 



or multiplying the last terms by (cos ^t)' + sin -x)-)^ = 1. 



[vw] = 2 sin ^Q' cos ^i)' (t'j^^fg ^2^^i ~ '^I'^'^i — ^2^''2) 

 — sin cos ^■ô- VW 

 = — 3 sin -0- cos ^d- VW . 



Per mutating we get the three relations 



[vw] = — 3 sin cos vw, 

 (115) [wu] = — 3 sin cos ^-ô- «w, 



[uv] = — 3 sin ^d- cos ^d- uv , 



