168 



(a, b) the coordinates of the point in which it intersects the perpendicular plane at the mirror; 

 ({) the focal length or interval between these two planes ; {«, /3) the cosines of the angles which 

 the near ray makes with the axes of (x) and (^), the given ray being the axis of (z) ; and 

 (A, B, C, D) coefficients calculated in [62.], which have not the same meanings here, as in 

 the four preceding paragraphs. These formulae give, by elimination of », the following- 

 biquadratic equation, 



F'./3*— 2;3i, [2(^ — AC) i^By—Cx) +{AD — BC) [Ay—Bx)\ Jf{Ay--Bxf=0, (G(8)) 



in vi\\\c\\F" = {AD — BC)t—^(B^ — AC){C'' — BD); when F" is negative, that is, when 

 the principal focus is inside the little ellipses of aberration [62], this biquadratic (GO) has two 

 of its roots real, and the other two imaginary ; but when F" is positive, that is, when the prin- 

 cipal focus is outside those ellipses, then the roots are either all real, or all imaginary ; so that 

 in the first case, any given point (x, y), near the focus, will have two rays passing through it ; 

 whereas, in the second case, it will either have four such rays or none. As these two cases are 

 thus essentially distinct, it will be convenient to consider them separately ; let us therefore 

 begin by investigating the density in the case where the principal focus is inside the little 

 ellipses of aberration. 



1st. Case. F" < 0. 



[72.] In this case, if we consider any rectangle upon the plane of aberration, having for its 

 four comers, 



1st. X, y; 2d. x-\-dx,y; 3d. x,y-\-dy; 4th. x -{- dx, y •\- dy; 



the rays that pass inside this little rectangle are diffused over two little parallellograms on the 

 perpendicular plane at the mirror, the corners of the one being 



. da , , db , 



] St. a,b, 2A. a -\- —T- . dx, * + -7- • dx, 

 dx dx 



da , , , db , _, da da 



b -f — — . dx 4. . du, and those of the other being composed in a similar manner of n', b' ; 



dx dy 



a, b, a', f, being the two points in which the two rays that pass through the point (x, y) are 



crossed by the perpendicular plane at the mirror. The areas of these little parallelograms, have 



for expressions 



fda db da db\ , , (da' db da' db' \ ^ ^ 



[dy zr-di d^h ^^-"^y' Vdi ^--dii^)' '^'- <V' 



and they are equal to one another, because 6' = — b, a' =z — a; also the area of the little 

 rectangle on the plane of aberration is dx.dy ; if then we denote by a "*> the density at the mirror, 

 the density at the point {x, y) will be nearly 



