60 Dr. E. H. Barton on a Graphical Method for 



have found its way into any of the ordinary text-books) the 

 labours both of students and demonstrators may be somewhat 

 lightened. 



Let the following notation and convention of signs be 

 adopted : — 



Focal length, /. 



Distance of object, u. 



Distance of image, v. 



All distances to be measured from the mirror or lens, and 

 to be reckoned positive when measured against the direction 

 of the incident light, and negative if with that direction. 



Then we have : — 

 For mirrors, 1 1 _ 1 ,... 



~vu-f' ■ S 1 ' 



and for lenses, 1 __ 1 _ 1 , 9 * 



v u f ' • • ' • \ / 



For the graphical method take two rectangular axes of 

 coordinates OX and OY, and let a straight line pass in any 

 direction through the point M„, whose coordinates are (/,/); 

 then it follows at once from (1) and elementary geometry 

 that its intercepts on the axes of a; and y give a pair of corre- 

 sponding values of u and v respectively for a concave mirror 

 of focal length /(see fig. 1). This is the case already referred 

 to as given in Aldis's • Geometrical Optics.' Now if the rota- 

 ting line pass through the point M*, whose coordinates are 

 (— /, — /), we have, as seen from (1) also, the case for a con- 

 vex mirror. Similarly, by consideration of (2), we see that 

 the points L a and L*, whose coordinates are (—/,/) and 

 (/, -/), give the completing cases for a concave and a convex 

 lens respectively. 



The first uses of this graphical method are to afford a 

 picture of the corresponding values of u and v for any mirror 

 or lens for which / is known, and to enable the student 

 to trace the series of values through which v passes as u 

 changes continuously from infinity to zero. But in this 

 view of the matter no very great advantage is gained, as it 

 is almost as troublesome to the student to remember the 

 positions of the fixed points through which the revolving line 

 must pass for the various cases as to recall and rightly use the 

 corresponding formulae. 



The second and more important use of this method is its 

 application to the determination of / for a given mirror or 

 lens when the optical bank is available with which to observe 



