Plummeu — On the Symmetrical Optical Instrument. 79 



This gives the locus of the primary focus. If p = p 2 where 



lip, = 2,,,-D 



the image becomes a straight line radial to the axis, for 



& - ?, + A& = 0. 



This gives the locus of the secondary focus. 

 Finally, if 



2 W (2C . ^-p" 1 + 2 N D- p-i.O 



p-i = 2 w (0+ 2?) = Kft _1 + P2 1 ) 

 it follows that 



>)/ - >)„ + Aij/ = CWV> & - £„ + ^£; = - 0>h%'- 



This gives the spherical locus of the circles of least confusion. If C = 

 astigmatism disappears, but curvature of the field remains. 



The quantities C and B are the pair most easily expressed. For, when 

 hi is omitted, 



W I Si K('\ 



and 



2 (if - KifTi \ jUj.j 2 /( ; 2 



v Lf {ftj ~ M-i) sir,, i \ i 



£ = SAW 2 



? 



^ /A 2 n ~ 2^-,^ + ^A',. Lln-^nLi+jMlQ 



2(Z l -JT i )«rA u^ 



h'i-i w" 



= s i>„a„ t 2^7- ''yfr » z « 2 '''(m« + fH-ii - W-i (2£j - Ki)\ 



*f*i fi-i ^i ~~ ly t) ' i 

 ^/.lifJ-i-lTi 



Hence the meaning of the Petzval condition is expressed by 



In the absence of astigmatism (7=0, p % = p„ = p, and this condition 

 secures flatness of field. But the independent condition for flatness of t In- 

 best possible field is 



= G'+I>=2 s ft^W |JT^,( w +^-i W i* 1 (2tf + 2JKifc-fffli. 



[18*j 



