49 



we find, finally, 



•d' being the same function as before. It results from this equation 

 (Q'") or from (M'") and (/'") that when the given ray is taken for 

 the axis of z we shall have 



, iYv (VV 3'F\ ST/J^v 3'»\1 . ,„, , 



if we put ^ = dx tan. 11, so that 11 denotes the angle which the 

 plane of projection makes with the plane of xz. Differentiating (R'") 

 for IT only, we find that the values of this angle which correspond to 

 the extreme positions of the focus by projection are determined by 

 the condition 



f^v d'F 3^y J«F\ 3»u/3^r S^F\ 3'F/S"« 3'u \ 



the planes of extreme projection, that is, the planes which correspond 

 to the extreme values of p, are therefore perpendicular to each other ; 

 and if we suppose them taken for the planes of xz, yz, and denote by 

 Pi, p, , the corresponding values of p, we shall have 



- '• • ■ , ,. i,0!;i!jv.;r. 



3«i; /3»F 



c" 3*u 3'F 3*1- ^"^ -" *»-• '*^'-' ^•" *'"• ' (^ ) 



;>i ~ 3^ 3% ~ 335 



-1 111 f I iJij "^ 



V 3%l3^ "*" 3^/'- / 



'v yy j/^ _ J»« 3'F __ 3«u 3»F i" 



3» 3x2 ' p, - 35^ 3%~ 1^ V ' -/ 



VOL. XVI. 



