584 Transactions of the Society. 



1-inch interval would not be perceived at 1000 yards, but at, say, 

 900 yards, in which case 



tan m — tan v = + o. 



We now come to the " order of merit " ; this is merely the 

 tangent of the visual angle divided by the tangent of the similar 

 instrumental angle, thus, in those instances where there is no 



" lag," when tan m = tan v, = 1 ; but, when the instrument 



° tan m 



was performing relatively better than unaided vision, would 



tan 7)i 



be greater than 1 ; and when worse less than 1. 



Lastly, to clear the table of decimals and initial decimal points, 



the quantities m and v were multiplied by one million ; also the 



formula for the " order of merit " was written 



tan m 



Further, it was pointed out that this last formula might with 



advantage be altered empirically to 



10 tan v 



tanwi — O.I. — t 



Where O.I. stands for the optical index, and t for the number of 

 thousands of lines to the inch resolved. Finally, the optical index 

 is the ratio of the numerical aperture to the initial magnifying 

 power, thus 



1000 N.A. 



O.I. = 



initial power 



For the benefit of non-mathematical readers, the value of 

 tan v is found by dividing the least interval separated by unaided 

 vision by the distance at which it is separated. (Note. — This 

 should be determined in very bright daylight.) 



Example : ^ inch can be separated at 45f inches, then tan v = 



' 2 _= 0-000437, this multiplied by one million = 437, the 



45-75 



value assigned to v in the following tables. 



To find tan m, multiply the magnifying power P (total com- 

 bined magnifying power when an eye-piece is used) by 100, and 

 divide by the member of thousands of lines resolved. 



Example : with 36 power, 8000 lines to the inch could be re- 

 solved, then, — — = 450 . 



The " lag" therefore is 450 - 437 = + 13. 



