THE GAUSS EQUATIONS FOR PARAXIAU RAYS. 19 
For the usual case in which object and image appear in the same medium 
(air), and/= /', the nodal planes coincide with the principal planes, which 
are then called the equivalent planes of any given lens system, as their dis- 
tance from the focal points is directly the equivalent focal length (E. F.)* of 
the lens system. 
THE FOCAL LENGTH OF A LENS SYSTEM OF TWO COMPONENTS. 
The microscope may be considered a lens system consisting of two com- 
ponents, the objective and the ocular, each one of which contributes its 
share to the magnification attained ; it may, however, be looked upon as a 
single system of definite E. F. with its focal planes in a definite position. To 
calculate the E. F. of the combined system, let L\ (Fig. 9), represent the 
objective lens and L z , the ocular; let F\ F'\ and f\, f'i be the foci and the 
focal lengths of L\, and F 2 F' 2 and / 2 , f'z, the foci and focal lengths of L^\ let 
the distance F'\F% = D (the optical tube length). Then the incident rays 
H\E, H Z G parallel with the axis pass through F'i, the posterior focus of LI, 
which in turn serves as object point for the lens Lz, and this in turn converges 
the emergent rays to F', the posterior principal focus of the combined system. 
In similar manner the rays FS and FR pass through F 2 , the anterior focus 
H, 
FIG. 9. 
of the lens Z.o, and emerge parallel with the axis ; the point F is consequently 
the anterior focus of the entire system. The rays H\E and H\S from the 
point HI are brought to focus by the system at H\, but as H\A = #'iA',the 
planes 7/i// 2 and #'i/7' 2 are the principal planes and the distances AF and 
*This abbreviation has been suggested by Dr. H. Kellncr as a concise expression for designating the 
distance of the focal point of a lens from its equivalent focal plane. This distance (aequivalente Brennweite) 
is one of the fundamental characteristics of a lens and determines at once the magnification produced by 
the lens under different conditions. The E. P. of a lens should not be confounded with its back focus 
(Schnittweite) or distance from the surface of the lens to its focal point. In place of the expression back 
focus, the term focus distance might well be substituted. 
