122 



2=-i. B.i.i'. (Q'O 



But we must not suppose that this area, like the area of the elliptic section (M"), is the entire 

 space over which all the intermediate rays, that is, all the rays making with the given ray angles 

 less than ( (), are diffused upon the plane of aberration ; for it is clear that these intermediate 

 rays intersect the plane of aberration partly inside the curve (P"J, and partly outside it ; since 

 the focus itself, that is the point y = 0, y' = 0, is outside that curve. We must therefore, in 

 order to find the whole space occupied by the intermediate rays, investigate the enveloppe of 

 all the curves similar to (P"), which can be formed by assigning different values to (6), and then 

 add to the area (2) of the curve (?") itself, the area of the additional space included between it 

 and its enveloppe. Differentiating therefore, the equation (?") for (t) as the only variable, we 

 find 



so that the enveloppe sought is a common parabola, having for equation, 



2C.i'.i/ = (AC — B').3f\ (R") 



and the additional space (2'), included between it and the curve which it envelopes, being equal 

 to the double of the definite integral 



-^.y{ Baf-c.^{iu-x'-) y. dx'. 



i.C i 

 taken from j/= 0, to x'= — ' ' , has for expression 



v(^ -f- C ) 



^'= ~i.fi. [-B + ^{B' + ny, 

 so that the whole space over which the intermediate rays are diffused, has for expression 



2 + 2' = i- i.fi. [ B + ^(B- + €')}. (S") 



In these calculations A, B, C, i, have been supposed positive: but the formula {S") holds 

 also when all or any of them are negative, provided that we then substitute their numeric for 

 their algebraic values. 



[61.] To find the geometrical meanings of the coefficients A, B, C, which enter into the 

 preceding expressions for the aberrations measured from a focus, let us investigate the curvatures 

 of the caustic surface. The two focal lengths of a ray, measured from the given perpendicular 

 surface, are determined by the formula (Q) of Section VI. 



da db 



(da \ f db\ 



d^- d* -^' 



, . , , y da da ^ db ^ db . , 



which when we make _ = _ j., -^ = 0, -^ = 0, -^ = — j„ j = {», gives by 



