10 



THEORY 



THE MICROSCOPE. 



any given system of rectangular co-ordinates, to which, also, can 

 be referred all other directions and points which come into con- 

 sideration. If we take the straight line in which the centres of 

 curvature lie as the abscissa-axis, and denote, for brevity, the 

 abscissae of the points N, Jf, N', M' by the same letters, so that 



FIG. 5. 



r = J/ N, r =M* N' (which values will therefore be positive 

 for convex refracting surfaces and negative for concave), the 

 equation for the incident ray assumes the form 



y = ( - 



+ If. 



(1) 



For readers who are little conversant with mathematical modes of 

 expression, we may add that y is the rectangular ordinate to the 

 optical axis, x the corresponding abscissa (taken from any given 



8 



point of origin), the tangent of the angle which the incident ray 

 ft 



makes with the direction of the axis, and 5 the ordinate of the 

 point in which the ray meets a plane drawn through N at right 

 angles to the axis. 



By the first refraction at P the ray takes another direction, 

 which will be determined by the equation 



y = (, - N*) + V, 



3' 



in which, as is easily seen, the quantities ' , and ft' are dependent 



71 



8 

 upon -Q- and 6, as well as upon the curvature of the lens. 



If the aperture of the lens is, as we assume, very small ill pro- 

 portion to the radii of curvature, the effective portion of the 

 refracting surface will nearly coincide with a tangential plane 



