Investigating Aberrations of Symmetrical Optical System. 497 



The construction of such formulae in the general case, 

 though doubtless the ideal aim of the method, leads of course 

 to results sufficiently unmanageable as to be of little value. 

 The present paper therefore considers only the case of chief 

 practical importance, namely, that of a symmetrical optical 

 instrument composed of media bounded by spherical inter- 

 faces with collinear centres in which only those rays are 

 regarded whose inclination 6 to the optic axis is small 

 enough to require the retention of low powers onty of 6. 



(1) Line- Coordinates. 



In space of three dimensions with the ordinary rectangular 

 Cartesian frame of reference, the equation to the straight 

 line whose direction cosines are I, m, n and which passes 

 through the point (a, ft, y) , is 



( X - x )/l = (y-/3)/m = (z-y)/n. . . . (1) 



Now these quantities a, /3, 7, I, m, n here employed to 

 define the straight line are, unfortunately, not unique for a 

 given line, since in place of (a, ft, 7) we may set the coordi- 

 nates of any other point on the line. The three quantities 

 /, m, ft, however, are organically connected with the line, 

 since they, and they only, can define its direction, but we 

 require further coordinates to distinguish our chosen line 

 from among the double infinitude of parallel lines having 

 the same direction (I, m, n). 



Reference to equation (1) shows that 



ny — mz = n/3 — m<y, h — nx=lry—na, m%~ly = ma.— l/3. 



That is to say, the three quantities 



ny — mz, Iz — nx, moc — ly 



are the same whatever point O, y, z) is chosen : in other 

 words they are invariants for the line. 

 We write 



a = ny — mz, 

 b = lz — nx, 

 c=mx-ly^ 



(2) 



and choose [I, m, n; a, b, c] as the coordinates of the straight 

 line. 



