60 



its ratio to the rectangle haib, is therefore expressed by 



~ [h: ib ~ u t^J' 



and by the equations (T*), or (S"^), 



fl tb ~ IS t^ ~ V' 



(U*) 



if we put 



M'l = (Aa + Bb) {Ca + Db) — [Ba + Cby . (V*) 



The smaller the quantity M" is, the more will the rays which 

 pass through the little rectangle ^a^b, be condensed at the principal 

 focus ; so that the curves upon the, plane of a, b, which have for 

 equation 



.¥" = const, (W) 



may be called lines of uniform condensation: and we see, by (F*), 

 that these curves will be ellipses or hyperbolas, according as Jsl" is 

 positive or negative, if we put for abridgment, 



(JB* — AC) (C* — BD) — 4 {AD — BC)* = N" . (X*) 



These elliptic or hyperbolic curves are all concentric and similar, 

 and their axes are all contained on the same pair of indefinite right 

 lines, which are perpendicular to each other and to the given ray ; 

 and the planes which pass through the ray, and through these axes 

 of the lines [W'*^), will coincide with the planes of a^z, yz, if the fol- 

 lowing condition be satisfied : 



AD — Be=zO, (Y*) 



that is, 



ifi^ }*^i0 J«3^» 



(Z*) 



