125 



{(A-C)i,' + (D-B)af+(BC—AD).i']\ 

 which may be thus written 



Ay* + 2 B"x'y' + C"^' — (D "y + £"x')« * + F'.«* = 0, (W" ) 



if we put for abridgment 



A" = (A — q'+4,B*, B" ={^A—C){D—B) — iBC, C"=(D — J5)» + 4CS 

 D' = {A+ C). B" + (-D + B). A", E<<zz{A + C).C" + (D + B).B", 

 F" = (AD — BC)* — 4!{B''—AC)iC^—BD). 

 These values give 



A"C" — jB"» = 4 {C(^ — C) + B(Z) — B)] '> 

 so that the curve (W") is an ellipse; the centre of this little ellipse has for coordinates 



a" = i(A + C).«^ b" = i(D + B).e, 



and its area is 



^ = ±i7r.{C.{C—A) + BJB — D)].6\ (X") 



If now we consider those intermediate rays, which make with the given ray some given small 

 angle (*'), less than {(), the points in which these rays cut the plane of aberration will form 

 another similar ellipse, having for equation 



Ay + 2B"x'i/ + C'af'' — (D"i/ + E"x') l'" + P'.i'* = ; 



and if (F") be negative, this ellipse is entirely inside the other, and all the rays that make with 

 the given ray angles not exceeding (6) are diffused over the elliptic area (X"). But if (F") be 

 positive, that is, if the focus be outside the little ellipse of aberration (W"), then the interme- 

 diate rays are not all diffused over the area of that ellipse, but cut the plane of aberration partly 

 inside that area and partly outside it. To find therefore, in this case, the whole space over 

 which these near rays are diffused, we must seek the enveloppe of all the little ellipses similar to 

 (W"), and then add to the area of that curve (W") itself, the area of the space included between 

 it and its enveloppe. This enveloppe has for equation 



(D"i/' + E"^Y = 4.F"-(^"y' + 2Byy' + C"x'^) ; (Y") 



when F" is negative it has no cKistence, and when F" is positive it consists of two right lines 

 passing through the focus, which are common tangents to all the little ellipses, and which may 

 be called the Limiting Lines of Aberration ; the space included between them and the ellipse 

 (W"), has for expression 



2'= — . (tan.^|.— ,J.) (Z") 



