October, 1903.] 



KNOWLEDGE. 



237 



the instrument, form an angle B with the optical axis. Then 

 we have, according to Abbe's sine-condition, 

 sine oL 



-■ — „ ^--- III. 

 sine 3 



But, according to Abbe's definitioD, sine a is equal to ii.a., 



therefore 



or, transposed, 



sin. 



sin. = : 



Bearing in mind that the pencils leaving the microscope are 

 always very slender ones, and p, therefore, so small an angle 

 that its sine and tangent may be considered as equal to each 

 other, we can say 



Radms of cone of rays 

 Distance to apex. 

 We had previously defined the /-value as the ratio 



Distance to apex 



Diameter of cone of rays 

 A comparison of these two ratios gives us at once 



I. f- value =^ -. — -■ = = ,-T—3 



J. sin. 3 n.a. 2 X n.a. 



or, in words — ;;( 



The f-ralue is found hy dividing the total iiiaijnification by 

 tirice the n.ii. 



We soon see that the rapidity is generally exceedingly small, 

 for, taking a typical example, say, ti.a. 7, magnification 350, we 

 get . , \iM 350 



/-value ^-- 2^^f^= ^.4 -^ 2o0 ; 



or, in photographic language, the microscope in this case works 

 at //250. 



Of coui'se, this low rapidity is the cause of the long exposures 

 required, and of the difficulty experienced in focussing the image 

 on ground glass. 



In the above argument we have shunned the question as to 

 what trigonometrical function is involved in this question of 

 rapidity, but if we stipulate that the /-values shall be such that 

 their squares are inversely proportional to the intensity of the 



illumination, the general formula used above, . ^ ^^ '«, 



sine P 



decides the question, for as all the light which the object sends 



into the objective is reunited in the image, but spreads over an 



area of »j-times the diameter, the intensity of illumination of 



the image is — ^ that of the object, or the /-value or the side 



of the image should be m-limes that on the side of the object 

 to co.iform to the desired proportionality, or in the same 



proportion as - _• Hence the /-value is or should be the 

 ' *^ sine •' 



reciprocal of twice the sine of the angle of convergence, and our 

 formula may be used for any convergence, although originally 

 derived for small angles. It follows that the /value of a 

 microscope-lens on the side of the object, or as used for micro- 

 photography, is correctly found by our formula = j, 



This formula shows that, from this point of view, microscope- 

 lenses are exceedingly rapid, for a low-power lens of n.a. -'ib 

 has a rapidity of //2, equal to the fastest portrait lenses. It 

 also shows that a dry lens {i.e., 7i.a. not more than 1) cannot 

 possibly be faster than /'/•5. 



We will finally consider the photographic lens used for 

 microscopical purposes and determine its n.n. from the known 

 /-value. The latter is always stated for the principal focus, 

 whilst we are going to use the lens at longer conjugates. If we 

 assume that the lens behaves like a thin lens of the same 

 e(£uivalent focus, the /-values will change proportionately to the 

 conjugates. For magnification X m these latter are respectively 



The resulting numerical aperture is, 



1 m 



m + 1 



{m+ l)/'and '""ti/-- 

 . . "' 



therefore, 



II. )i.a.^= 



-/-value 



2 X (to + 1) X/-value 



fication by the product of twice the /-value and one plus the 

 magnification. 



Thus a lens working at /;6 and magnifying, say, four times 

 has a numerical aperture 



4 4 1 



= 1 ' ?- - ,^ - -067. 



■> X .) X 1, t)IJ 1.0 



The formulae I. and 11. solve the problems defined in the 

 introduction to this article in every possible case ; they can 

 even be used in the case of immersion-objectives, for the above 

 deductions apply to these lenses if we substitute n. sine a for 

 sine a, n being the refractive index of the immersion-fluid. 



or, in words — 



The numerical aperture of a photographic lens used for 

 photomicrographical purposes is found by dividing the magui- 



THE COLLECTION, EXAMINATION, AND PRESER- 

 VATION OF MITES FOUND IN FRESH 

 WATER {IlydrachnidcB). 

 By Ch.\s. D. Soar, f.r.m.s. 

 {Continued from yage 212.) 



On reaching home the contents of the bottle should be emptied 

 into a porcelain dish such as photographers use, when it will be 

 noticed that the mites generally swim in the corners or along 

 the sides, and can then be removed with a pipette to a large 

 tube filled with clean water in which some AimchtiriK is placed. 

 This latter will keep the water clean and fresh for a considerable 

 time. 



Experience will dictate which species can safely be kept 

 together, a matter in which some discrimination is required, 

 because some varieties prey on others, such, for instance, as 

 Limnenia on Eulais. 



Undoubtedly the best plan is to proceed with the examination 

 at once, because a great part of the brdliancy of colouring is 

 lost in a short time, and the mites are much more lively when 

 freshly caught than subsequently. I have, however, kept mites 

 alive in a tube 4 in. by 1 in. by adding fresh water to rejdace 

 that evaporated, for a period of twelve months. 



The best method of examination is to place the mite on a 

 3 in. by 1 in. glass .slip, turning the specimen on the ventral or 

 dorsal side as may be re(iuired, and having every part extended. 

 A cover-glass is then laid over the specimen, and sufficient clean 

 water is allowed to flow between the cover-glass and the slip to 

 fill the intervening space. The specimen may move its limbs 

 and palpi for a short time, but it soon becomes quite passive, 

 the weight of the cover-glass being tufticient to retain the body 

 of the mite in position. The slip is then laid on a piece of 

 white card on the stage of the microscope, and illuminated by 

 reflected light; a U in. objective will usually be found the 

 most suitable. 



The advantage of this arrangement is that the specimen can 

 be reversed, and both sides examined, and by having an aperture 

 in the cardboard, a further examination may be made by trans- 

 mitted light. In this latter condition the hairs and claws can 

 be seen very distinctly, particularly if the light be thrown a 

 little obliquely. After examination the specimens can be 

 returned to the tube, and are u>ually none the worse. 



To preserve the specimens they should be placed in the 

 following solution : — 



10 parts glycerine 

 10 „ distilled water 

 3 „ citric acid 

 3 ,, pure spirit. 



They can be placed in the s-olution alive, and although at first 

 the limbs will be contracted, they subsequently retract. It 

 also preserves the colours of hard-skinned mites fairly well. 



If at any time it is desired to make a mounted preparation of 

 any mites preserved in this way, ihey can be transferred to 

 cells containing the same solution. If required for balsam 

 mounts, the ghceriue can be removed by repeated soaking in 

 absolute alcohol, subsequently passing them tlirough clove oil. 



It will be found that balsam-mounted specimens will have a 

 tendency to vaporize ; this can be obviated bj- making a small 

 hole in the body of the mite in a position which is of no con- 

 sequence, and thus allow the balsam to penetrate. 1 think the 

 soft-skinned mites mount best in glycerine solution ; I do not 

 mount in this medium myself, but have some beautiful 

 preparations by Mr. Taverner, ia which the construction is 

 shown to the best advantage. They have been in my possession 

 for some time, and show no signs of deterioration. 



