16 THEORY OF THE MICROSCOPE. 



Similarly, we obtain for f*, if we put this value in the equa- 

 tion (13), 



If, therefore, we suppose that there are two points E and U* on 

 the optic axis, whose abscisste (which we also denote by E and 

 E*) are equal to the values of f and *, which we have just found, 

 and if we draw through them two planes at right angles to the 

 axis, then the first will be met by the incident ray at a distance 

 equal to that at which the second will be met by the emergent ray . 

 The points in question are, therefore, no other than the principal 

 points of the system, and the planes drawn through them are the 

 pi^incipal planes. 



It may also be easily proved that if n = n* a ray directed to 

 E will emerge without deviation. For, if, in the equation for the 

 incident ray, that is, in 



y = (x -NO) + W, 



IV 



x is made equal to E, and for (E N) its value from (14) is 

 substituted, then, since y will be 0, we get 



consequently 



If we introduce this value in the expression for fi* (equation 10), 



8* 8 

 then /3* = /3, and, (since n* = n), *= 5 > tnat ^ s > tne emergent 



ray forms the same angle with the axis as the incident ray. The 

 ray directed to E is a ray of direction. 



It is not necessary here to follow out further the consequences 

 which result for the case of n and n* being unequal, and therefore 



8* 8 



^-i and A being to one another as n to n*. 



n* n 



As the second case, let the position of P and P* be such that 

 the image-point is at an infinite distance, and consequently 



