112 



and considering the quantities W, U, as functions of s, which we 

 shall suppose capable of being developed according to positive inte- 

 ger powers of that variable. In this manner we shall obtain 



du . du du „ du 



d» at dfi dt ^ 



and therefore by (D), 

 We have also by (E), 



t.= ^_,.^TTr7, ^ = ^+|i. (T') 



in which 



(U') 



r = v'l — «; 

 and we may put 



dW wA „ ,. „, ... 



-dT = 2 -^ ^^if^ (* + 2) '"^ "'•^■+* ' 



"ST - 2 — f^^i — + = -(,■) UiM «■+' ; 



,,_,., o, , 1.3.5...(2t 4- 1) z 



c^+i = (» + 2) «.^^ + -2X6:.w + 2T-2- ' 



in which WW , W2i + 4 , ^, are constants of the same kind as be- 

 fore, A denoting the distance of the origin from the focus of central 

 rays. Hence, if we put for abridgment, 



. - -■' ^' + f fV'^ 



