TELESCOPE. 



As we explained how Table I. is dcriwd from an uni- 

 verfal dioptric theorem, we (hall explain how the theorems 

 in Table II. are deduced from one fundamental equation, 

 on a fuppofition that the fines of incidence and refraaion m 

 glafs are always as 3 : 2. Let L N, in fg. 6, reprefent 

 a convex lens, Of its axis, O a radiant point therein, O A 

 a ray proceedinjr from thence to A, a point m the furface 

 LBN; then if C is the centre of convexity of that furface, 

 C G, drawn through the point A, will be perpendicular to 

 that furface in the point A ; C A or C B is the radius. A/ 

 the refraaed ray, and / the point where it meets the axis 

 after the firft refraaion. Let D B ; f/, C A = R, E B 

 = /, the thicknefs of the lens ; and let the fine of the angle 

 of incidence A G be called m, and the fine of the angle 

 of refraaion C A/ or GAH be called n. Now, fince 

 the point A is luppofed to be very near to the vertex B, 

 OA may be confidered equal to OB = D, and in the 

 triangle C A O, we (hall have A O to AC as the angle C 

 to the angle O ; that is, rf : R :: C : O. Alfo O B + B C 

 = D + R will be as the oppofite angle CAO or O AG, 

 the fines of both being the fame. Then as m ; n :: rf + r : 



^ " "*" ^ " , which will be as the angle C A/; this, taken 

 m 



from the angle ACQ = d, leaves the angle A/O 



= D'"- R" - D". Laftly, as the angle/ : O :: A O 

 m 



Dm — Rn — Dn 



we multiply the equalioii by t, md add thereto rm, we fliall 

 have dm -^ rm — dn 



DRmi — Tibtb + Rnth + Dbrm — Rnmr 



or OB : A/ or B/; that is, as 

 RDm 



R 



D 



Dm - Rn — D» 



= B/, the diftance of the point/ 



in the axis, after the firft refraaion. But fince there is a 

 fecond furface L E N of the lens, there muft neceffarily be a 

 fecond refraaion of the ray A O to fome other point in the 

 axis, as F, in f?. 7. In this cafe, the refraaion being out 

 of a denfe into a rare medium, the fine of incidence will 

 be to that of refraaion the rcoerfe of what it was before, 

 v'tz. as n to m ; that is, the fine of I af is to the fine of 

 I a F as B to m, which, in the cafe of fingle refraaion, was 

 as m to n. Here let K a be called r, and E/ — d ; then 

 there will be ^ : ,• :: K : /, and E/ + E K = af + r, 

 which will be as the angle /a K, or its complement I a/; 



therefore n : m ;: </ + r : , which will exprefs 



n 



the angle I a F. Then I a F - a K F = '^"' "^ ''"' -d- 



rm — dn 



= aKF. Now, as F : K :: Ka or 



KB : a F or E F ; that is, as • : d :: r : 



n 



^-^—=EF. EutB/-BE=^ ^^'" ^ 

 dm+ rm — dn ■' D»;— R« — D« 



— t =. d = E/; therefore, putting m — ff = 3, we fhall 



, , DRm URm-'Dbt + rnt 



have a = =r-r -— — t — — 



D3 - Rb 



Alfo dnr — 



DR. 



D3 - R« 

 Y> btnr + rtnnr 



D5- R« 



Again, dm -f rm — dn — D* + rm; if, therefore, 



D4 - R» 



Then 



dm 



dm + rm - 

 T) Kmnr 



dn 

 - Hbttir + Rtnnr 



= EF- 



DRmb - Dbtb T Rntb t I>brm — Rnmr 

 This laft equation may be abridged, by fubllituting p for 



-, , that is, for , then we (hall have 



b m — n 



= EF. 



pDRmr — p Dblr J- Rtnrp 

 DRm - Di/+ Rnt + Tirn — pRmr 



Laftly, if we take n ■=. ph vn p'Qblr; and m — n = 3 in 

 Tibt \ this equation will be finally reduced to this funda- 

 mental equation, wi. 



pYiRrm — Dtrn + Rtrpn VV— f 



D Rm — Dim + D/n+ Rtn + 'Drm—pRrm~ ~^' 



The ratio of «i to n being taken in glafs as 3 : 2, we (hall 



have- 



» 3-2 

 equation will then ftand thus ; w'a. 

 6DRr - 2D<r + 4tR'- 



2 =j» for a glafs lens, and th# 



3DR-3Dr-D<+2R*-6Rr~^^ ~^' 



and when t, the thicknefs, is difregarded, we have from this 

 fundamental theorem all the various theorems contained ia 

 Table II. for finding the geometrical focus under all the 

 various circumftances that are likely to occur in the pofition 

 of a fingle lens, where the refraftive power is not adverted to. 

 To illuftrate the refpeaive ufes of the theorems con- 

 tained in the two preceding tables, we muft fuppofe the 

 ratio between the fines of the angle of incidence and of re- 

 fraaion known by fome of the ufual modes of determining it 

 experimentally ; and then, when the ratio of m : » is fo 



determined, there will be = a, the fymbol intro- 



n 



duced in the theorems of the firft table ; when d is equal to 

 the diftance of the radiant, r the radius, and / the proper 

 focus determined by real refraaion through the glafs ufed, 



dr — rf 



2df 



For inftance, Martin ground a piece of white flint-glafs 

 with a tool of 21.5 inches radius, into a double convex lens, 

 and when a lamp was placed at the diftance of 417.25 inches, 

 the refraaed focus was meafured accurately, and found to 

 be only 18.75 inches; whence, according to the theorem, 



the theorem for finding the value of a is 



we have 4i7-^5 x 21.5 + 21.5 x .8.75 

 2 X 417.25 X 18.75 



= 0.599 = a 



1-599 

 I 



; and if we put n — 1, then m will be 1.599, ^°'" 

 ■ = 0.599. When the fun is the radiant, then 



d becomes infinite, and the theorem becomes, as in the firft 

 f 2 X. 7 



table, — , which gives, in this cafe, ^ =. 17.94 for the 



2 d 1 . 1 90 



refraficd 



