Asti<^inatic Aberration. 525 



namely, iu those experiments in which the aberration is 

 wholly or in part asticrmatic. In investigating the properties 

 of non-homocentric pencils or beams with one or more kinds 

 of aberration present, it is necessary to determine what part 

 is played by each of the several kinds of aberration. It is 

 of interest to know what is the effect of, or part played by, 

 astigmatic aberration in tlie production of shadows by objects 

 placed in, or moved about in. non-homocentric pencils or beams. 

 It will be found that the presence of astigmatic aberration in 

 a non-homocentric beam is accountable merely for the twist 

 or rotation shown in the shadow or image formed by an 

 object placed in or moved about in the beam. 



To examine mathematically the twists produced in a pure 

 astigmatic non-homocentric pencil or beam, the symmetrical 

 standard pencil, such as is investigated by Heath (Geome- 

 trical Optics, p. 142), is assumed. 



If the axis of the beam is the axis of :, and the origin is 

 at the intersection of this axis with a selected orthogonal 

 surface or plane cutting the beam iu an ellipse of which the 

 axes are 2 :z, 2 /3, respectively, the equation to the bounding 

 surface of the beam is 



where t'l and r^ are the distances of the focal lines from the 

 selected plane. 



If 0, 77, are the coordinates of a point in the selected 

 ellipse it is easily shown that 



^= and 1] = 



and if the eccentric angle of the point (^ ij ()) in the ellipse is 

 (j) then 



^= a cos (^. 



77 = /3 sin 6, 



:. = .(l-^^)cosc^. 



Heath, Geom. Optics, § 142, p. 161. 



