14 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
THE GAUSS EQUATIONS FOR PARAXIAL RAYS. 
REFRACTION AT A SINGLE SPHERICAL SURFACE. 
Light-waves travel more slowly in glass than in air, and if incident on a 
plane glass surface their direction of propagation is deflected according to 
the sine law sin a = ri sin a', where a = angle of incidence measured from 
the normal to the plate, a' = angle of refraction; = refractive index of 
first medium, '= refractive index of second medium. In applying this 
formula to the refraction at the spherical surface 5 (Fig. 2), of a single lens, 
let n be the refractive index of the first medium and n' that of the second 
medium. Assuming that the rays travel from left to right, we find that a 
ray from M incident at B is deflected along BM' while the central ray MAM' 
passes through the spherical surface without deflection. From the triangles 
M BC and M'BC it is evident that 
sin o MC , sin a' M'C 
_ _ 
sin <f> MB sin <f> BM' 
Hence, by division 
sin a n'MC BM' 
= ^ 
sin a~ n~ M'C BM 
For small-slope angles u, u' we may substitute, as a first approximation, 
MA for MB and M'A for M'B and obtain the equation 
MC M'An' x-r x n 
M'C' MA~ n x'-r x n 
which on rearrangement becomes 
n(x r) n'(x' r) n n' n n' 
-L= -- _ L or --- = _ 
. 
x x x x r 
if the origin of coordinates be at A, and MA = x, M'A = x r , AC = r. 
The action of the lens is therefore to convi-r^r to the axial point M' waves 
of light emerging from the axial point M. The relation between the points 
M and M' is reciprocal; they are said to be conjugate points or foci. 
For a second point near the axis (Fig. 3) and at a distance from C equal 
to CM, equation (i) can be applied directly and the point />' on the axis 
PCP' located. Assuming the distance pM to be small, we may substitute 
