808 Transactions of the Society. 



to enlarge the image witbout displaying new details of the objects, 

 the increase of the di-sipation-effects with increasing v is not a 

 serious drawback, because the visual angle of the minutest detail 

 is increased in the same proportion. Though the absolute pre- 

 cision of the image will be diminished under the higher v, the 

 relative will remain the same, and must still be sufficient for the 

 more enlarged image if it was sufficient for the less enlarged. 

 Now as tvAce that super-amplification which raises the defects of 

 the systems just up to the threshold of vision, is always borne 

 by objectives without any considerable or objectionable loss of 

 definition, the increase of the ocular-power alone will be sutTicient 

 for reaching the upper limit of generally useful amplifications 

 for the various apertures which is shown by the fifth column of 

 Table I. Nevertheless it may be desirable that such higher 

 powers should be occasionally obtainable under still more favourable 

 conditions, that is with the lowest figure of v. I admit, therefore, 

 that objectives of shorter focal lengths — down to A a//" the values 

 assigned above at the utmost — may still be useful for the immer- 

 s/o/i-apertures. (Not for dry lenses, because it would be decidedly 

 irrational to force such high amplification from apertures which 

 leave the delineating power much below the attainable limit.) 



On the other hand, it is quite certain that even the minimum 

 powers which are required for the utilization of the said apertures, 

 may be obtained under much higher super-amplification with a 

 sufficiently satisfactory quality of the image, when the utmost 

 degree of perfection is not required or when objects of less sensi- 

 bility, e. g. diatoms, are in question. Twice the critical value of v 

 (i. e. 8, 8, 12, for the three systems respectively) will, however, be 

 the limit in regard to objectives intended for somewhat general 

 application, and 2i (i.e. 10, 10, 15 respectively) the utmost ad- 

 missible figure in regard to lenses for diatom work (the minimum 

 amplifications, 480, 640, 720, being regarded) ; because if still 

 higher ocular-powers should be requu-ed even for these minimum 

 amplifications the deterioration of the image, attendant upon the 

 enlargement of the aberration -circles, will become so perceptible, 

 even with the least sensitive objects, that satisfactory recognition of 

 the minutest details must be unquestionably lost. Though the 

 details which are within the reach of the aperture may still be 

 seen, the quality of the image will be so much inferior to that 

 obtained by higher objective-powers and lower ocular-powers that 

 it is obviously unwise to obtain under unfavourable conditions what 

 may as easily be otherwise obtained. I must therefore consider as 

 irrational constructions all those wide-angled lenses which do not 

 yield even the lowest total power required for proper utilization of 

 the aperture, except by a stdl greater amount of eye-piecing than 

 is assigned above. 



