VOL. XXIV.] PHILOSOPHICAL TRANSACTIONS. 187 



object still less than 4-r : then will Inddr — ndrr and Inddr — drr be negative 

 quantities; oy ndrr — Inddr and drr — Inddr will be positive quantities. And 

 here, if n be greater than 1, then will ndrr — Inddr be greater than drr — 

 Inddr ^ or f greater thany^ But if n be less than 1, then will ndrr — Inddr be 

 less than drr — Inddr, or f less than f. That is, in the concave speculum, if 

 the distance of the object be less than half the radius of the speculum, then 

 while the object recedes from the speculum, the image recedes from it also ; or 

 while the object approaches towards the speculum, the image approaches to it 

 also. 



And all these conchisions, which we have deduced from the steps of the cal- 

 culation, are contained in one scholium in Dr. Gregory's Catoptrics. 



'Coro/.' 4.^— In the equation /== -^——, if 4 be supposed infinite, it will be 



f •=. ^r\ which is a rule for parallel rays, or for a radiating object placed at an 

 infinite distance. The same thing will follow, if h be made infinite in the 



. ^ rr •>(■ rb 



equation/= ^q:^- 



" lK)ro/. '5. —In the equation /= -^-^, changing the negative sign of the 



quantity r into positive, it will be / = . '^ ; or in the equation f = — --^-r 



changing the positive sign into negative, it will bey* = ~ ^ ; which gives a 



rule for a speculum convex towards the radiating object. The change of the 

 sign is very plain ; for as in the concave speculum it is rf = Z> + '*> so in the 

 convex it will be rf = /> — r. 



Corol. 6. — In a convex speculum, (those things continuing which are stated 

 in corol. 3, concerning the concave speculum) it will appear, that if n be greater 

 than 1 , then Irndd -j- ndrr will be greater than Imdd + drr ; and if n be less 

 than 1, then Irndd -}- ndrr will be less than 2rndd + drr. That is, both the 

 object and the image at the same time either approach towards the speculum^ or 

 both recede from it. 



In a convex speculum it also appears, that if the object recede to an immense 

 distance, yet its image will not recede from the vertex more than half the radius, 

 but that it will there stop, in the middle point between the centre and the ver- 



tex : for supposing either d or b to be infinite, it will bey = ^ or -r, that 



is, in either case y = Jr. 



To these may also be added the solution of a catoptric problem, viz. To 

 find such a position of the radiating point, in respect of a given speculum, that 



B B 2 ; 



