_ _9 



124 Transactions of the Society. 



actual lenses which could be measured by a foot rule, but it is 



the distance between the equivalent lenses, and it is this distance 



which may appropriately be termed the natural optical tube length. 



When two lenses are in optical contact (tig. 18), the focus of the 



combination is determined by dividing the product of 



•piO 10 ff 



their focal lengths by their sum ; thus, F = *'., where 



F is the focus of the combination and/,/' the foci of 



the plano-convex and the meniscus. For example, in 



fig. 18, if / the focal length of one lens, is 2, and /' 



2x3 6 

 that of the other, 3, then F = ^ -3 = 5 ; = H- When, 



however, the equivalent lenses are separated by a distance d, which we 

 have called the natural optical tube length, it is necessary to subtract 

 this quantity from the sum of the foci in the denominator. For 

 example, let the same two lenses be separated by a distance 1, then 



F = H- = 2 X 8 = H.* It must be pointed out that, in 



f + f'-d 2 + 3-1 2 . 



this case, F is positive and the image is erect ; but if the distance d, 

 the natural optical tube length, be increased so that it is greater than 

 the sum of the foci, F will become negative, which indicates that the 



2x3 

 image will be inverted. Example: — let d = 8, then F = 9 , ._> _ o : 



This is the condition which prevails in the compound Microscope. 



One more little piece of elementary arithmetic, viz. that p, the 

 combined magnifying power, is determined from the focus by dividing 

 the conventional quantity 10 by the focus ; thus, in the last example, 



p as — _ = — 5, the negative sign indicating the inverted image as 



before ; conversely, if the magnifying power is known, the focus F 



can be found ; thus F =- - = —^ — - 2. So far for the natural 



p — o 



optical tube length for the present; but we have another optical 

 tube length, viz. one which was introduced by Prof. Abbe, and 

 which may be called the conventional optical tube length A, to dis- 

 tinguish it from the one we have just been considering. Prof. Abbe's 

 tube length is the distance measured between the foci. Fig. 19 

 illustrates the two kinds of optical tube length, where the natural 

 optical tube length d is measured from e to E, and the conventional 

 A from / to F ; the plano-convex lens in the figure represents the 

 lens that is equivalent to the objective, but the eye-piece has both 

 lenses drawn. It will be noticed at once that A is equal to d less 

 the sum of the foci (e f+EF); in other words, A is the denominator 



. ff ff 



in our fraction above, viz. F = - • / -;, ^ = =^-r. 



J + J — a — A 



• This formula is given by Coddiugton (1830), part ii. p. 34. 



