ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 827 



Fiually, the focal poiuts can be deterrained correct to some tenths of 

 a millimetre, by the aid of the sun's rays without the use of a Microscope, 

 in the following way : — The diopter under examination is fixed with wax 

 to the slide of a slide-rule furnished with a vernier reading to tenths of 

 a millimetre, and the optic axis is brought parallel to the rule. A piece 

 of black paper is gummed on the rule at the zero of the graduation, and 

 the surface of the diopter is brought in contact with it ; a first reading 

 is taken, and then, with optic axis turned towards the sun, the diopter is 

 separated from the paper until the observer, with the aid of a lens, sees 

 the refracted cone reduced to a brilliant point. A second reading is taken, 

 and the difference gives the distance of the focus from the first refracting 

 face. 



Microscopes are usually provided with a set of eye-pieces and objec- 

 tives, which can be associated in different ways. To calculate in 

 advance the dioptric power of any associated system, we have the formula 

 for a binary system of diopters Fj, F2 



^ = clF,F,-(F,+ ¥,), 



where d is the distance between the second principal point of the first 

 diopter and the first principal point of the second. If e^ e^ and E denote 

 the thickness of objective, eye-piece, and whole Microscope respectively, 

 T^i' the distance of the second principal point of the objective to its last 

 face, and r/^ the distance of the first principal point of the eye-piece to 

 its first face 



d = 'E - (ei + e.) + Vi + Vz- 



Just as the dioptric power of the eye-piece was obtained from those 

 of the objective and of the Microscope complete, so by an analogous 

 method can its cardinal points be obtained. 



Thus supposing known 



El, e„ e,. 



7/1 r]i the distances of the principal points of the objective to the 



corresponding refracting surfaces. 

 4> the dioptric power of the Microscope. 

 Fj „ „ objective. 



F2 „ „ eye-piece. 



p',p the distance of the second focal point of the Microscope to 



the last refracting surface. 

 We have, to determine 772? d = E — (ej -t- ej) + rji + rj^, 



but 



a> + F, + F2_ $ ,/,/• 



Therefore -q-^ is known, and by adding algebraically the focal length 

 /j the distance of the first focal point from the first face is obtained : 



To determine the second principal point, denoting byj^'c/) and q'^ 

 the distances of the second focal point of the Microscope from the last 

 refracting surface and from the second focal point of the eye-piece 



