18 C. V. L. Charlier 



12. The transition from the »old» to the »new» coordinates can be suitably 

 performed in 5 steps, when instead of /, m, n polarcoordinates are introduced by 

 the formulae 



I = sin 5 cos A, 



(23) m = sin 5 sin A, 



n = cos B . 



These 5 steps are the following: 



1. b is exchanged for d'-, 



2. and ^ire exchanged for B and A ■ 



3. B, A, u^,v^, 'U\; u^,v.^, iv,^ are exchanged for B, A, u^' , v^' , tv^' ; v^' , tv^'; 



4. B, A are exchanged for 'j;'; 



5. \>' is exchanged for b'. 



The determinant 7 is thus split up into 5 factors according to the scheme 



(24) I-l{t]xi{^^ l]xih: -V "Ox/(j'f)x/P;). 



Consider in order each one of these transformations. 

 I first call attention to the well known theorem 



^25^ J A^i ' ^2 ' • • ■ ' j = ]^ . J I ^1 ' ^2 ' • • • ' 



V^l) ^2' ^'V X"*^! > "^2' '^'V 



which is immediately found through performing first the direct transformation and 

 then the inverse one. 



'3- 



According to the formula (12) we have 



and hence 



db _ Ä;V, + m,) _J_ _ k'\m, + 

 Ô7~a*~ (o^ cos^d ^i^i^g ^^j, 



or 



0-/ /(;^(w2^ + »"a) '"^ 



14 if*' 

 We have 



B,Aj ' ']> 



