506 Dr. G. J. Stonoy on Microscopic Vision. 



between the dotted lines. Then a, 8, and 8 ! are the three 

 perpendiculars of one of the triangles of fig. 1, both in actual 

 magnitude and in position. Now the lengths of the perpen- 

 diculars of a triangle are inversely as its sides. Take then 

 the reciprocals of cr, 8, and 8' without changing their positions 

 and 111 



1 gj g( a a ) °) C 7 



and are in the positions of the three perpendiculars of fig. 1. 

 Draw the triangle of fig. 2 with its three sides parallel to the 

 perpendiculars of fig. 1. Its sides will then be proportional to 

 a, b, and c. Again 8=\ / /sina=\/g ; and 8 ; '=%/ /sin a! = \/g f : 

 therefore ^ 



-, g, g' a a, b, c 



and are in the positions of the three sides of fig. 2, in addi- 

 tion to being represented in magnitude by the lengths of 

 those sides. Observe that X/«r, g, and g' are numbers, i. e. 

 their scalar parts are of cypher dimensions. Now if p and p' 

 of fig. 3 are the puncta in image x of the two beams, their 

 radii represent g and g f both in position and in magnitude 

 estimated on the X scale, which means that scale which 

 prevails throughout image x, and on which the length of the 

 radius of image x represents G, the grasp or numerical 

 aperture of the objective. The X scale is one in which 

 lengths mean numbers. We thus learn that the triangle in 

 fig. 3 is in the same position as the triangle in fig. 2, and is 

 similar to it. Therefore d, the line joining the puncta p and 

 p', represents X/cr in magnitude on the X scale, and represents 

 it also in position. In other words, 



I C 

 <tt=z\ — d, or more simply <r=X/d, ... (8) 



if we identify G and R. Hence a is equal to X divided 

 by the number represented by d on the JL scale ; and, further, 

 the ruling of which a is the spacing has its luminous bars 

 perpendicular to the line d. 



This is a very important proposition, far reaching in the use 

 that can be made of it in the interpretation of microscopical 

 phenomena. We should, however, when employing it, 

 bear this caution in mind — In every attempt to draw an 

 inference from image x, we must recollect that the inform- 

 ation it gives, though great, is limited. It tells us the 

 intensities and the positions of the beams of uniform plane 

 waves into which the light is resolved — beams which are 

 thrown off from the whole extent of the objective field ; it 

 also tells us the directions and the spacings of the rulings 

 produced by these beams ; but it is silent in regard to every- 



