Tracing Caustic Curves. 265 



k its radius ; F''T is the longitudinal aberration (usually 

 denoted by a) and F"L' is the lateral aberration denoted by I. 

 Now in the books it is said that F"D may be taken as \a 

 or }F"T, and k as ±1. In this case the distance F"D of the 

 Least Circle from F" is about l-F'T and k is about j?l. 

 This example is sufficient to show that this simple rule 

 cannot be relied upon when a lens is used with its full 

 aperture. In most cases it will be sufficient to find fP when 

 <f>' is midway between and its extreme limit and the value 

 of p at the point/; K, the intersection . of the extreme ray 

 with the small arc drawn through /, will enable one to deter- 

 mine the size and position of the Least Circle with a high 

 degree of accuracy. 



In the diagram H /X H represents the second Principal Plane 

 cutting the extreme ray at H, and the angle H"TL or DTK 

 is yfr or <f>' — <f>, i.e. —48° 57' 32". Indeed the angles DTF 

 and H/'TH are geometrically equal to i/r, though they are 

 measured in the reverse direction. 



Clearly from the diagram 



(1) a or F"T = F / 'H"-TH".-=F ,/ H"-H"HcotH' / TH 



=f"—y cot yjr, where y=H"H. 



(2) I or F' 7 L' = TF" tan F"TL'=a tan yfr. 



(3) k or DK = TD tan DTK = TD tan &. 



Now (2) and (3) give the correct numerical result which 

 is all that. is required for I and k, for they are essentially 

 signless ; in (1), however, it is necessary to know whether 

 a is positive or negative. A moment's consideration will 

 show that (1) is universally true when the symbols carry 

 signs of direction, for as H"H is negative and IP'TH is 

 positive, every term in the equation carries the same (nega- 

 tive) sign. 



In order then to determine a and I it is only necessary to 

 trace the extreme ray, but to find the size and position of 

 the Least Circle it is necessary to trace that part of the 

 caustic which the extreme ray cuts. 



The caustic in fig. 4 is extraordinarily large as it even 

 extends beyond the Principal Plane, although the aperture 

 of the lens is relatively small. A much smaller caustic is 

 formed by the same lens if its position is reversed as in 

 fig. S, where the semiaperture of the lens is far greater. 

 If, however, the source of light be at or near the first prin- 

 cipal focus of the planoconvex, so that the incident light is a 

 widely divergent cone, this position of the lens with its plane 

 surface facing the incident light gives far the smaller caustic. 



