Vol. IX] WOODWORTH— OPTICS OF THE MICROSCOPE 193 



the same as those given above, omitting the primes and ending 

 at F, which is the principal focus for paraxial rays. The 

 distance FG is therefore the spherical aberration. 



The correctness of this method of calculation is proven as 

 follows : In the triangle CDL', L'DE is by construction = x 

 and since CD/CL' = n/n' the angle DL'C = a'. Since ^ 

 equals zero and the external angle L'DE = CL'D + DCL' 

 we have from the equation a • — 9 = at.' — o' ; DCL= e' 

 and therefore CL is parallel with J'K'. In the same way, in the 

 next refraction CM' can be shown to be parallel with K'G. 



In the calculation of the paraxial ray it is at once evident 

 that a ray half way between I and J would by this method 

 of construction proceed to a point half way between H and K 

 and then exactly to F. The same would be true of a ray ^4 oi" 

 Ys above I and H, that is the distance from the optical axis 

 does not affect the focal distance F; therefore a paraxial ray 

 comes to a focus at this point. 



The methods above described constitute a complete scheme 

 for lens calculation which can be carried out grafically with 

 as much accuracy as is required for practical purposes, since 

 it is well within the limits of the accuracy of the physical data 

 and mathematically to any degree required for theoretical 

 investigation. 



IV. On the Aberration of Depth 



The aberrations due to the thickness or depth of objects 

 have received very scant attention though it is well known 

 through observation, as well as from the theoretical considera- 

 tions, that there are such aberrations. 



The figure accompanying this article (Fig. 7) illustrates 

 the amount and character of such aberrations in the simple 

 case of a single lens surface, and of an object limited to a 

 single axial plane. 



It illustrates at the same time an application of the methods 

 of graphic calculation and the new oblique axis method of 

 calculation, which is available either for graphic or mathe- 

 matical computations. The "air curve" and "glass curve" 

 are drawn in the manner already described about the "graph 

 center" with radii proportional to the index of air and of 



