K.\ AMPLE AFTER GAUSS III 



The Gauss system of tracing a ray through two or more lenses 

 on an axis illustrated by means of a worked- out example. 



Two lenses, 1 and 2, fig. 87, or an axis ./ // are given. No. 1 is 

 ;i double convex of crown ^ inch thick, the refractive index p. being 

 :;. the radius of the surface A i> ;| and that of B 1 inch. No. 2 lens 

 i> a ] ilano-concave of flint y 1 ,, inch thick, the refractive index /z being 

 ?, the radius of the Miriace (' i> , ;ind the surface I) is plane. The 

 distance between the lenses, that is> from B to C measured on the 

 axis, is | inch. The problem is to firtd the conjugate focus of any 

 given point V. 



In order to accomplish this two points have first to be found with 

 regard to each lens. These points are called principal points (see 

 PP', QQ' in fig. 87). When the radii of curvature r and r', d, 

 the thickness, and p l /* 2 , the refractive indices of the respective 

 lenses, 1 are known, the distance of these points from the vertices, i.e. 

 the points where the axis cuts the surfaces of the lens, can be found. 

 Thus by applying Professor Fuller's formulae to lens 1 the distance of 

 P from the vertex A can be determined seep. 115 (i) similarly P' 

 from B p. 115 (ii). In the same way the points QQ' from C and I) 

 in lens 2 can be measured off (v) (vi), pp. 115, 116. 



It must be particularly noticed that in measuring off any dis- 

 tance if the number is + it must be measured from left to right, 

 and if from right to left. Thus in (i) p. 115 because the sign of 

 158 is + P lies '158 of an inch to the right of A. And in (ii) 

 because '21 is P' lies -21 of an inch to the left of B. The same 

 rule applies to the radii ; thus the radius of A, being measured from 

 the vertex to the centre or from left to right, is + ; but the radius 

 of B, being measured from the vertex to its centre or from right to 

 left, is . Similarly with the concave surface, C being measured 

 from right to left is . 



In both the examples before us the points PP', QQ' fall inside 



any number multiplied by = 0. cx > plus, or minus, or multiplied by any number is 



still cc. 



The following are the rules for the treatment of algebraical signs : 



In the multiplication or division of like signs the result is always plus; but if 



the signs are dissimilar it is always minus. 



In addition, add all the terms together that have a plus sign ; then all the terms 



with a minus sign ; subtract the less from the greater and affix the sign of the 



greater. Example : 



+ 3-4 + 2-5= -4. 



In subtraction change the sign of the term to be subtracted and then add in 

 accordance with the previous rule. Example : 



-3 

 + 2 

 -5 



An example occurs in the annexed equations (x) and (xi), p. 116, of -* = + r 

 but then the + is changed into a - by the negative sign in front of the fraction. 

 In (xii), p. 116, however, there being a + in front of the fraction, the result remains- 

 positive. 



1 In the worked-out example no distinction has been made between the r, r 1 of 

 one lens and the r, r' of the other lens, as well as of /j. and d, because when the 

 principal points and focal length are determined for one lens those expressions are 

 not again needed, so the same letters with different values assigned to them may be 

 equally well used for the next lens. Too manj' different terms are apt to confuse 

 the student, while those who are familiar with mathematical expressions will under- 

 stand the arrangement. 



