52 C. V. L. Charlier 



Passing on to terms of a higher order we consider 



According to (18) we have, for = , 



vw 



II ' = u. sin + Mo cos ^■9- — cos sin \ — cos — -\- sin <h ^ 



Vj' = sin ^d' -\- cos ^d- — cos è sin ^ i cos 4» ^"^^ -= + sin (jj . 



\ y 11^ -\- v'^ \/ u- -{- v'y 



= w^ sin -\- w^ cos ^■fl' — cos ^ sin !)■ ( — sin j/».^ w") 



"2' ^ 4~ ^2 ^'^ ~l~ ^^^^ ^ ( — 'I' -7=== + sin 4» 



\ y 4- v'^ 



etc. 



Putting 



(110) 

 so that 



a;^ cos jLj , —j== = AJj^ SJ u iy^ ; 



= a;, cos , ^ — = Sin Z„ 



and 



= sin ^0- + «2 cos ^t)- -|- cos ^ sin ■& l/y'^ + ?-(;^ cos (4 + L^), 

 ^1' ^1 ^i^^ ^2 + *^ ^'^^ ^ Vw'' + cos (4 + Lj), 



'^1' ^ "'1 ~'~'^*'2 "^'^ "i" ^ ^^^^ ^ Vu'^ + cos (4 + L3), 



««2' = cos ^■9- + «2 sill — cos & sin & ]/ + cos (4 + , 

 ^2' ^ ■^i ^^2 s'n — cos ■& sin d 'I ^^m;^ + u'' cos (4 + Lo), 



w^' cos ^■9' +W2 sin ^t)' — cos sin {)■ V'^^r- + r- cos (4 + L3), 



where = 270". It is not necessary to write down the values of and . 

 We now get 



Mj'' = (m^ sin + «2 cos ^'ô-)'' + 



+ ( (?/.! sin ^-9 + M2 cos 2*)-'-i (y2 + w^)'/= cos -9 sin & cos (4 + + 



+ [ J (Mj sin 2^ + «2 cos 2^)'-2 (y2 + cos sin cos ^(^ + L^] + 



+ ) (m^ sin 2^ + cos ^*)'-3 (v^ + w^)'/^ cos sin cos ^((j; + 4- 



