( -87 )

r —=z but is the two Convexities be os the fame Sphere, so as r = &

rf {

then will the Distance be = p r, that is, is the Lens be Glaß — z r, so that ifan Object be placed at the Diameter os the Sphere distant, in this case theFocus will be as far within as the Object is without, and the Species repre-sented thereby will be as big as the Life; but if it were a Plano-Convex,the fame distance will be = if r, or in Glaß, to four times the Radius osthe Convexity.

A second Use is to find what Convexity or Concavity is required, to makea västly distant Object be represented at a given Focus, aster the one Sursaceos the Lens is formed - } which is but a Corollary of our Theorem for finding f,having />, d, r, and f, given ; for d being Infinite, that Rule becomes

rf

— x, that is in Glaß

rf

— ?, whence if / be greater than 2. > ,

pr-f " *r-f

r' f

f becomes Negative, and — - -is the Radius of the Concave sought.

But to return to our first Theorem , which accounting for the thicknefi ofthe Lens, we will here again resume, vi%. 'mpdr% — nd%t -p- nprft

m dr -+■ mdf — mp i r — m — hdt -+■ nrt

And let it be requir’d to find the Focus where a whole Sphere will col-

lecb the Beams procecding from an Object at the distance d. Here t is equal

to zr , and r — x; And aster due Reduction, the Theorem will stand thus,

mpdr — zndr ■+ Znprr r ,-s J . 7C - •• n.j

£ ---— f: but u d be lnhmte, it is contracted to

i sj + i»r — mp rZ n — m

r = fy wherefore a Sphere of Glaß collects the Sun's

tnp r

z n Zm — Z n

Beams at half the Semidiameter of the Sphere without it, and a Sphere ofA Vater at a whole Semidiameter. But if the Ratio of Refraction m to » beas z to 1, the Focus falis on the opposite Sursace of the Sphere, but if it beof greater Inequality it falis within.

Another Example shall be when a Hemisphere is exposed to Parallel Rays,that is, d and x being Infinite, and t = r, and aster due Reduction, the The-

orem

resolts

n n

m m

-r — f \ that is, in Glaß it is at -f r, in Water

at | r, but if the Hemilphere were Diamant, it would collect the Beams

at r? of the Radius beyond the Center.

Bb i

Lastly,