i8 
METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
From the similar triangles MQF, ABF, and H'A'F' in Fig. 6 and from 
equation (8) we obtain the equations 
(9) 
These two equations (8) and (9) are simple and fundamental, and express 
the behavior of the central rays in any given lens system. They can be 
derived in a number of different ways and are applicable to all lens systems 
with focal points. As noted in the derivation they are strictly valid only for 
central, paraxial rays. 
Substituting the values from (7) and (9) in 50, we have 
/. 
r 
do) 
In a lens system the refractive index of the first medium (object space) is 
often equal to that of the last medium (image space), and for this case 
/= /'. The principal focal lengths are equal and of opposite sign. 
PRINCIPAL PLANES. 
y' 
The two conjugate planes for which the lateral magnification /3 = = i are 
y 
located at x=f and x'= /' (equation 9), or at distances FA and F'A' 
(Fig. 7) from the focal points equal to the principal focal lengths ; they are 
called the principal or Gauss planes and are useful in representing graphi- 
cally the effect of an entire lens system consisting of any number of compo- 
nent lenses. 
NODAL PLANES. 
The two conjugate axial planes for which the angular magnification 
y = = i are called the nodal planes. Their position is readily found from 
tan H 
equations (50), (9), and ( 10) to be x = -/' and x' = /. (Fig. 8, N and N') 
