12 THEORY OF THE MICROSCOPE. 



sin a' TI 

 or, since - - = 7 (according to the law of refraction), 



iy . ^'1*, M oQ. (5) 



n sin 



From the equations (1) and (2) for the incident and refracted ray 

 we get, however, for the ordinates of the points Q and (/, for 

 which x N = r Q , the values 



y ( 



If we substitute these in equation (5), then 



/3>o __ ^ sinj> , 0,0 



^' ' n'sin 0' T 5T/ 



or 



sn 

 "r "sin 0' r 



In this expression the first term on the right-hand side is strictly 

 accurate ; but, since </> and <', according to our supposition, differ 

 from a right angle only by small quantities, and, therefore, their 

 sines differ from unity by quantities which, as compared with the 

 latter, are very small, it follows (since the quotient must in 

 addition to this be multiplied by the small quantities 6 and /3) 



that 



w' ^ 

 P = /3 - ,, - V, (6) 



which is accurate for quantities up to the third approximation. 



It is, therefore, clear, without further explanation, that if the 

 equation for the path of the ray after the second refraction at 

 the posterior surface of the lens is brought into the form 



y = J (x - N*) + !>*, (7) 



the quantity ft* can be determined through ft' in exactly the same 

 way as the latter is determined through ft . We have, therefore, 

 only to change the accents in the expression just found for ft'. It 

 then becomes 



