169 



and it remains to calculate the coefficient in the second member. For this purpose, I observe 

 that in general, when any four quantities a, b, x, y, are connected by two relations, so that 

 a, b, are functions of x, y, and reciprocally, their partial differentials are connected by the 

 following relation, 



(da db da db\ f dy dx dy dx\ 



Hy' 1^~ ~dx'liy) ^~da"db~~db''da)~^' ^ ^ 



it is sufficient therefore to calculate -^' -rz ^•"j~* Now, the equations (F*)) give 



^C- dx^(Aa + Bb). da + (Ba + Cb). db 

 ^^\ dy= (Ba + Cb). da + (Ca + Db). db 



♦■<•■ (f • ^ - ^ ^) = («"+ «)■- 1-^" + ^*Xc. + «« 



=:i\[{B» + Cfi)^—(A» + B/i)(Cu + Dfi)]; 



and if we put (S« + C/3)* — (A» + 5/3)(C» + D/i) =: M, we have by the same equations 



M.» = (Ba -\- Cli). y — (C» + Dfi). 

 M.(3 = (Bcc + C/$}. x — {A» + Bfi). 



we have therefore 



M.CC = (B» + C/i). y — (C» + Dfi).x l , j^^g,. 



It results from this expression, that when the principal focus is inside the little ellipses of 

 aberration considered in [62], there exists another remarkable series of ellipses upon the plane 

 of aberration, determined by the equation 



M = const. (MW) 



and possessing this characteristic property that along every such ellipse the density of the re- 

 flected light is constant. The ellipses of this series (M(8)) are all concentric and similar, having 

 their common centre at the principal focus, and having their axes situated on two remarkable 

 lines, which are perpendicular to each other and to the given ray, and form with that ray three 

 natural axes of coordinates passing through the principal focus. 



[73.] Suppose then that we have taken for our axes of coordinates, the three natural axes 

 just mentioned, the ray from which the aberrations are to be measured being still the axis of «; 

 we shall have the relation 



AD — BCz^O, {N(8)) 



and the expression for the density at a point (r, v) upon the plane of aberration will become 



