117 



(T"'), and by then comparing the corresponding powers of r, that is, of the aberration at tlic 

 caustic surface. Thus, if we make «=1, the equations (T"") become identical; if« = 2, they 

 become 



. _d)l_ d'-a dy^ d^ ^1/ / da y t^y rfg J^ ^'£K. ( Jl. ^' 



that is, 





2 sin. V = 



<^y 



to 



as in the formulae (1), (2), or (U'") of [68.] ; if s = 3, they become after reductions. 



dx' d'a /d'x' d^a dW _^\ db ■' d^x' 



da ' dVr^ "^ \da.db ' dVr^ "^ db^ ' dVrV ' dVr/ db^ 



^ ^ _ j;' / db \^ 



that is "' - " 



= 6.-^. .■+S. (j;^ . 2. + -jj^ . 2rf). »+-j^ . »-, 



as in the formulae (1)', (2)' of the same paragraph ; s = 4 gives 



dx' d^a . . f d'x' d'a . d^x' £b_\ db_ 



= 



^ \da.db dVr^^ 



da ' dVr* ' \da.db d^r^^ db^ ' d^/r^J d^r 



"*■ Wa.rf6^ * «/v'r^ + db^ ' dVi^l \d^r) ^ df ' \d->/r) ' 

 4. ^ / "^V / i^a y „ <f'3^' _£«_ _££_ . ££_l^bX\ 



