119 



axis of (z), and the tangent planes to the two developable pencUs passing through it for the 

 planes of (x, z), (y, z) : we shall then have, [40], 



« = 0, ^ = 0, y=l, a = 0, 6 = 0, c = 0, X = 0, Y=.0,Z=^, 



dc^O, d(=:0, dZ=:0, dX = da + idcc, dY z= db -\- ^d^, 



4,, {jj, being the focal lengths of the mirror; and substituting these values for the partial dif- 

 ferential coefficients of X, Y, in the general expressions (K") for the lateral aberrations, we 

 find 



{ being the distance from the mirror to the plane on which the aberration is measured. These 

 formulae (L") are only the equations (E') of the IXth section, under another form ; and it fol- 

 lows from the principles of that section, that the whole lateral aberration may be thus expressed, 



(■^ \/{u^ + /3/) being the angle which the near ray makes with the given ray ; (u) a constant 

 coefficient, depending on the position of the near ray, and determined by the equation 



M = ({i — J2). sin. L. cos. Z,. 



(L being the angle which the plane of (x, 2) makes with a plane drawn through the given parallel 

 to the near ray, so that /3, = a,, tan. /,) : and 



5 = 5 — ({,. COS. 'L + gg. sin. ^L) 



being the distance of the point where the aberration is measured from the point at which that 

 aberration is least. It follows also, that if we consider any small parcel of the near rays, the 

 area on the plane of aberration over which these rays are diffused, is equal to the product of the 

 distances of that plane from the two foci of the given ray, multiplied by a constant quantity de- 

 pending on the nature of the parcel. If, for instance, we consider only those rays which make 

 with the given ray angles not exceeding some small given angle {6), these rays are diffused over 

 the area of an ellipse, having for equation 



VOL. XV. 5 



