22 C. V. L. Cliarlier 



Hence we get 



(B, A\ _ k' + (T3 - C cos i?)^ 

 \a'>, '];/ sin »ßsiuD 



or if 'Cj, ê and C are expressed in polarcoordinates: 



I 



B, Ä\ sin sin cos -<{' + (cos ß — cos ■& cos BY 



'^1 sin ^/^ sin \> 



This expression is, however, simpHfied with the help of the sphaerical triangle 

 fig. 4. Denoting the angle S[— Line of symmetry) by rp, we got 



sin ß cos (j; = sin 1^ sin tp 



cos ß = cos B cos ^ -\- sin B sin ■0- cos y 



so that 



(30) /(^'^^ 



>>, ']> / sin 5 



15. I 



I'll^, f^, '^-('f, U^. V^, W,y 



\u^, V^', IV^'\ u^', 



We observe that B and A (as well as I, m, w), whicli determine the position 

 of the line of symmetry, are invariant parameters in this substitution. 



The value of this determinant may be found rather directly in the following 

 manner. 



We have 



, , «(;^ ; , d^, , j \u^, w^; u^, Vj, w. 



It is, however, shown in Medd. 69 that, if m/, y/, w/; u^', Vg', Wg' denote the 

 velocities before the passage, then the velocities after the passage have the values 

 v^, Mg, v^, W.2- Analytically expressed we have 



2m^l , — 



= -j (.Mg — / + — i;^ Hi + IV. ^ tl), 



and inversely 



2m^l , ^ 



u„ = u„ [u„ — H, i + — V, m + — n), 



^ ^ -\- ^ 112 1 12 1 / 



/ ^ , ^ -, ; X 



u. = li. -\ — m„ — u. ( + — t', m + — iv, n), 



^ ^ mj + ^ ^ ^ ^ 2 ^ 



2m^ I 



Ug — w,' (m,' — u.' I 4- v/ — v/ m 4- wJ — w/ n) 



^ - w, + W2 ^ 1121 12 1 / 



which expressions directly show, that 

 ^?<i, I'-j, ?<2i ^2' ^ 



Mg'^ î^2 ' ^2 / V'^i' *^2' ^2' ^3 



^ ?<,, I'-j, ?<2. ^2- ^2 \ ' ^'i ' "^1 ; "2 ' «^2 1 ^2 



