in 



r ^ r, and from d = — v', to » = + v', »/ being the semiangle between the limiting 

 lines, and consequently 



tan, 



-' = J^=Ji-^ m 



we find for the whole number of the near rays that pass within a given distance r from the 

 focus ♦ 



// A («). rdrdv = 2 A n^^^.r T, , (UW) 



and, therefore, for the density at the focus itself as compared with that at another point of the 

 same kind, 



A(f)= A(f*Vr,, (V(8)) 



r, denoting the transcendental 



"^' ~fo V\(C^—BD). COS. ^v—iAC—B"). sin. ^v] " (^'*'') 



[75.] The preceding expressions may be put under other forms, some of which are simpler. 

 Thus, if we still suppose the axes of coordinates to coincide with the natural axes determined by 

 the equation (N^*)), so that the axis of the reflected system may be the axis of z, and the 



common transverse axis of the lines of uniform density the axis of x; if also we denote by 



a: 



a" a"v/-1 

 the density of a point upon this latter axis, and by or the density of a point 



upon the axis of y ; we shall have by (O^*') (S®) the following approximate expressions for the 

 density at any other point upon the plane of aberration, 



r, V being the polar coordinates of the point, and e the excentricity of the ellipses or hyperbolas 

 at which the density is constant ; and the formulae (Q'^>) (V(*)) for the density at the principal 

 focus become 



A(*) = A'./? ± ) 



■^ o (1— e'. sin. ^v)i ^ 



•^ <, (1— e"-. sin.'^«)t J 



e being less than unity in the first and greater in the second. With respect to the value of 

 this excentricity, it is equal to the cosecant of the imaginary or real angle v' determined by the 

 formula (T(^^); it is also connected with the position of the ellipses of aberration [62.], by this 



VOL, XV. D D 



