130 Transactions of the Society. 



Putting this value of a in the equation ahove, we have — = — -f- 



- = , and a = f (m + 1). This last expression is all we 



should require if we could measure a, the distance of the ground-glass 

 screen from the equivalent lens, but this is just what we are unable 

 to do, because we do not know the position of the equivalent lens ; 

 we can, however, measure the distance of the object from its image, 

 and this distance, viz. a + a, we have made 100 in. Now, when 

 the image of an object is projected to a distance which is consider- 

 able in comparison to the focal length of the lens projecting it, the 

 object will be situated very close to the principal focus of the lens, 

 and therefore we may write f for a, without introducing any great 

 error. 



We have then a +/ = 100, and a = f (in + 1) ; 



therefore / (m + 1) + / = 100 



100 



and / = 



m + 2 



The error of which we have just been speaking increases as the 



focal length of the objective increases, but is practically of no moment 



provided that the distance of the object from the image remains 



100 in. ; as we have in the Table no objective with as much as 3 in. 



of focus, it might be as well to examine what the error will amount 



to if a 3 in. objective is calculated by the above formula. Let m = 



100 

 31-3, then/ = 7^T~h = 3*0. If the focal length is calculated by a 



longer though strictly accurate formula, / = 2 '99713; the error 

 therefore is only + 0-00287. If a higher power is taken, the error 



will be less. Let m = 48, then / = _ = 2*0; the longer 



4o —J— Jj 



formula makes / = 1-99917 ; the error is + - 00083, and quite 

 insensible. 



The formula in column B is derived from this one ; for as the 



initial power is equal to — , it is therefore equal to We 



may therefore infer that in using these simple and convenient 

 formulae no error worthy of our consideratiou has been introduced 

 into the Table. 



In fig. 19, the objective and the curves of the lenses of the eye- 

 piece are diagrammatic, but the position of the lenses of the eye-piece, 

 as well as the surfaces E E' and the foci F F' of its equivalent lens. 

 are correctly placed. It will be seen that the front surface E of the 

 equivalent lens is behind the back surface E', but the front focus F is 



