5Q6 PHILOSOPHICAL TRANSACTIONS. [anNO 16Q3. 



But to turn to our first theorem, which accounting for the thickness of the 

 lens, we will here again resume, viz. 



— "'Pdrf-ndft + nprH ^ ^_ ^^^ j^^ j^ ^^ ;^^j ^^ ^^^ ^^^ 



tn d r + m a f — mprf — m — n. dt + nrt "^ ^ 



focus, where a whole sphere will collect the beams proceeding from an object at 

 the distance d: here t is equal to 2 r, and r =: j : and, after due reduction, the 

 theorem will stand thus, ^^^ *"!" " — '_ri_pr_r _ r. y^^^ jj- ^ j^^ infinite, it is 



2nrf+2nr — m;>r "^ ' 



contracted to — ^ — r = ^ r := f ; wherefore a sphere of glass col- 



lects the sun's beams at half the semidiameter of the sphere without it, and a 

 sphere of water at a whole semidiameter. But if the ratio of refraction m to w, 

 be as 2 to 1, the focus falls on the opposite surface of the sphere, and if it be of 

 greater inequality, it falls within. 



Another example shall be, when a hemisphere is exposed to parallel rays, 

 that is d and j being infinite, and t ■=■ r; then after due reduction the theorem 



results r = f. That is, in glass it is at 4- r, in water at Z- r ; but if 



mm — mn -^ ' o j 



the hemisphere were diamond, it would collect the beams at -^V of the radius 

 beyond the centre. 



Lastly, as to the effect of turning the two sides of a lens towards an object ; 

 it is evident, that if the thickness of the lens be very small, so as that you 

 neglect it, or account < = O ; then in all cases the focus of the same lens, to 

 whatever beams, will be the same, without any difference on the turning the 

 lens : but if you are so nice as to consider the thickness, (which is seldom 

 worth accounting for) in the case of parallel rays falling on a plano-convex 

 of glass, if the plain side be towards the object, t occasions no difference, 

 but the focal distance /"= 1 r : but when the convex side is towiards the object, 

 it is contracted to 2 r — ^t, so that the focus is nearer by f /. If the lens be 

 double convex, the difference is less ; if a meniscus, greater. If the convexity 

 on both sides be equal, the focal length is about -^ t shorter than when t = O. 

 In a meniscus, the concave side towards the object increases the focal length, 

 but the convex towards the object diminishes it. A general rule for the differ- 

 ence arising on turning > the lens, where the focus is affirmative, is this, 



^~, for double convexes of differing spheres. But for a meniscus, th^ 



2 r t 



3r + 3f - I 



same difference becomes ^'^^ t^^^ i of which I need give no other demon- 



3 r — 3 f + t 

 stration, but that by a due reduction it will so follow from what is premised, as 

 will the theorems for all sorts of problems relating to the foci of optic glasses. 



