1903.] The Theory of Symmetrical Optical Objectives. 269 



Before applying these results to our special problem it is necessary 

 to show that / <7 = is equivalent to Petzval's condition for Flatness 

 of Field. 



From equations (II) it is evident that 



FIS - GIS Fi2 - Gi2 F 23 - G 23 F - G 



C 13 2 C 12 2 



from (III) 



Hence if/- g = 0, we have 



hat is, using I, we have 



Z/lfr-i)}.* or S {(I-1)1}-0, 

 Ui2 l\ fh/r J 



which is Petzval's form. 



It is evident that, if this condition be satisfied for a single system, 

 it is also satisfied for the double symmetrical system and vice versa. 



Application to the case of Double Symmetrical Objectives. 



Consider the object plane at (0), the image due to the combined 

 system at (3), the stop at (2), and let (1) be the plane symmetrical to 

 (3) with regard to the stop. 



Let 



+ 4F 12X i 2 2 + 2Gi 2 pi V 

 then from symmetry 



T 23 = A 12/ 3 3 2 + B 12/ > 2 2 + 2Ci 2X 23 + Di2/> 3 4 + E 12 /) 2 4 + 4F 12X 23 2 



let 



TOI EE A i/>o 2 + . . ., 



where A i = B i = C i = o7 -, and 



assuming %i = 1. 



VOL. LXXII. 



