186 JPHILOSOPHICAL TRANSACTIONS. , [aNNO 1705. 



Corot. 1. — First, if d be greater than r; then, by the calculation,/, or 

 ._ ^ , is always greater than ^r. That is, if the distance of the radiating 



point be greater than the radius of the speculum, then the focal distance will 



dr 

 always be greater than half that radius. Also _ will always be less than 



r; that is, the focal distance always less than the said radius of the speculum. 



Secondly, if d be == r, then will /= ^^^ = r: that ip, if the radiating 



point be placed in the centre of the speculum, its image *ill there be united 

 with it. 



Thirdly, if d be less than r, then the expression for / vyill be either positive, 

 or negative, or infinite, according as the quantity 2d is either greater than, less 

 than, or equal to the quantity r. If id be greater than r, that is d greater 

 than 4r ; then the focus and the radiating point lie on the same side of the 

 speculum. But if 2d be less than r, or d less than -^r ; then the image will be 

 in the axis of the speculum produced beyond the vertex. And if 2d be = r, 

 or d =z 4-r ; then the image is at an infinite, dkUn«;^ or the reflected ray be- 

 comes parallel to the axis. ufi^ fiufanim.^ fyt.n \ 



Coral. a.-r-By means of this calculation it may be readily determined, how 

 the motion of the image corresponds with the motion of the radiating object, in 

 respect of the speculum. Let the distance of the image from the speculum 



dr 

 be __ as before, when the distance of the object is d. Now let the distance 



of the object be any how changed, and from d let it become nd, making n any 



dr 



nomhier, integer or fraction : then instead of the former equation f = „, ^ 



•f- ^ ■' • ndr ''• • 



we shall have ^ == . _ , another equation to a new focus. Where, when 



n is greater than 1, the second distance of the object is greater than the former; 

 tut less when n is less than 1 . 

 - This being premised ; if <£ be greater than r, and n greater than ] ; then will 



F be less than /I that is -~ — ^ less than -3 — , or 2nddr —r ndrr less than 

 •'' 2nd—r 2d — r 



^r^r^^"^ drr, as is manifest. That is, in a concave speculum, if the distance 

 of the object be greater than the radius, then while the object recedes from the 

 speculum, the image approaches towards the same. But if it be less than I, 

 then 2nddr — ndrr will be greater than 2nddr — drr, or p greater than /* 

 That is, as the object approaches toward the speculum, the image recedes 

 from it :i) touomiB- jo miit 



Suppose now that d is less than ^r ; and let nd be any other distance of the 



