TELESCOPE. 



in practice, when r and R are unequal, from our former 

 practical theorems, thus : let us determine tlie compound 



F X F 



focus of B and D by tlie theorem =,, as they have 



F -t- F 



each a pofitive focus, and call this focus = geometrical, 



and then — will be $ refra&ed ; and fecondly, let 'F geo- 



'F 

 metrical be turned into — - for the rcfrafted focus ot the 

 2 b 



F X 'F 

 concave C ; and tlien, by the theorem — — = = *, we 



F — F 



fhall have * for the compound refracted focus of the triple 

 objeft-glafs, or length of the telefcope : for example, taking 



'•47'5 

 1.524 



= -9655 X T = A of the fubftituted lens E, with 



the fame data as before, we ihall firft have 



10 X 20 

 10 + 20 



30 



= 6.66 = t?, and -^—z, or — = 6.28 = <p refrafted : 

 1. 06 2 a 



and 



10 /'F\ 



— [—,) = ii-33 = 'F refrafted: alfo, by the 



(p X 'F 

 theorem j=^ , becaufe 'F has a negative focus, we get 



6.28 X 8.33 ;2-33 . , 



;:; ^r^ = - = 2 ?.!:■? = *, as was required. 



8.33 — 6.28 2.05 -^ ^ 



We (hall ufe this method of finding the compound focus (p 

 of two lenfes, and alfo *, the focus arifing from all the 

 three lenfes, where it is to be underftcod that a, the fymbol 

 for the refraftive power of both the crown lenfes B and D, 

 is taken the fame ; and that we exprefs the refradtive power 

 of C, tlie concave, by the fymbol l>, as a fubftitute for 'a. 



We mull further premife, that when T, the thicknefs of 

 each lens B and D, is not confidered, we fhall fliew pre- 

 fently that the fpherical aberration, arifing from any fingle 

 lens that receives the rays of light, may be diminifhed as 

 4 : I, by the combination of two lenfes, to be fubftituted 

 tor that one. Bearing thefe premifes in mind, we now pro- 

 ceed to the confideration of a triple objeft-glafs, that fhall 

 have the due corrections for both the prifmatic and fpherical 

 aberrations. It will greatly facihtate both our explanation 

 and exemplification, if we fuppoie the two convex lenfes 

 B and D, having a compound focus ^^ ^, to be reprefented 

 by a fingle lens E, with the fame focus <p, but with a Jimi- 

 nijhed aberration ; for then we may proceed nearly as in our 

 feven preceding examples ; but reverfing the procefs, when 

 the concave has its radii given, to find the convex lens. 



Example 8. — Let it be required to conftrudl a triple 

 achromatic objeft-glafs of 30 inches focal length, with the 

 fame refraftive and difperfive pov.»ers as in the firft example ; 

 •ui%. with m : n in the crown as 1.528 : I, and in the flint 

 as 1.5735 : I ; and with the difperfive powers as i : 1.524; 

 and let the two radii of the concave be each 13.9, fo as to 

 have 'F, as in the firft example. 



In the firft place we have V : 'R :: i : I, and, as we have 

 feenabove,'A ;= 1.666 x 'T,by the general theorem of Huy- 

 gens ; in the next place, becaufe the conca-veC is given to find the 

 convex E, the corredting number z, found as before, becomes 

 a muhiplier, in a revcrfed operation, and we have 1.666 ('A) 

 X 'T X .883 (z) — 1.4715 X T ='A correP.ed: we mult 

 alfo ufe the former multiplier 1.524 (the difperfive power, 

 or proportional focus) as a divijor, and then we get 



a focus = ; but there is no fuch fmall quantity of aberra- 

 tion in any one lens. Let us however fee what the abfolutc 

 aberration 'A will be, unconncfted with the fador 'T, which 

 faftor we iiave determined, from the verfed fines to the radii 

 13.9 and 13.9, to be z= .1653 ; therefore 1.4715 x .1653 

 = .2432 =; 'A abfolutcly. Now we have fecn, in the firil 

 example, that .252 is = T of the proper convex ; let ua 



now confider that — ^ = . 1 26 is = T in one of the lenfe» 



2 



B and D, whicli we propofc to make in every refpcA fimi- 

 lar, in order to have as few different curves, and confequently 

 as few different tools, as poffiblc ; then, becaufe .243a is 



the ahfolule aberration of C, tlie concave, we have ^ 



.126 

 = 1.93 X T = A of either of the convex lenfes; but T 

 is .126; therefore 1.93 x .126 = .2432 is the abjelule 

 aberration of each convex lens, exactly equal to the abfo- 

 lute aberration of the concave. But we have alTcrted, and 

 fhall demonftrate hereafter, that when the thicknefs of the 

 lenfes is neglcfted, a proper combination of two lenfes, 

 placed at a certain diftaiicc from each other, will diminifh the 

 aberration belonging to one four limes, and even when the 

 diftance = o, this will be nearly the cafe : now we hare 

 .9655 X T = A in the lens E of equal focus, let us leave 

 out A, and multiply by 4, and we have 4 x .9655 = 3.862 

 very nearly, the fum of the aberrations (without T) of the 

 two convex lenfes B and D, taken together ; viz. 2 x 1.93 

 =: 3.86 ; but yet the abfolutc aberration of each feparate 

 convex lens (T being confidered) is exaftly equal to the 

 abfolute aberration ('T confidered) of the double concave. 

 This relation of the refpeftive aberrations being once efta- 

 bhfhed and confirmed by praftice, which Tullcy affirms to 

 be the cafe, fimplifies llie complex bufinefs of calculating a 

 triple objcft-glals : for \.\\c fum of the abfolute aberrations of 

 the two convex lenfes of like glafs, muft be always equal to 

 double the abfolute correded aberration of the concave 

 alone, in order to have a due corre£tion for fpherical aber- 

 ration, and confequent indiftinctnefs. Hence, when the fo- 

 cal diftance, ?, of the two convex lenfes B and D is in tlic 

 fame proportion to 'F, the focal diftance of the concave, 

 that tlieir feparate difperfive powers are relatively to that of 

 the concave B, the relative radii of either B or D, or of 

 both, may be "uar/Vi/ at pleafurc, provided that the_/i/m of 

 their abfolute aberrations remain equal to double the abfolutc 

 aberration of the concave C, and provided that ^, their 

 compound focus, be not altered. But we have not yet ad- 

 jufted the two focal diftances fo as to make the object -glafs 

 achromatic, and *, or the compound focus of the three, to 

 be equal 30 inches. From an equation of the aberration 

 1.93 X T, or from .2432 abfolutely^ we find the ratio of 

 r : R in each lens to be as 1.34 : i, which is alfo agreeable 

 to Tulley's tables, from which this ratio may be had by in- 

 fpedtion ; alfo the rational geometrical focus for thefe num- 

 bers is 1. 145. Now according to our firfl example, wc have 

 feen that when 'F = 13.9 in a telefcope of 30 inches focal 

 length, F will be 9.12, when there is only one double con- 

 vex lens ; but here we have two lenfes to produce 9.12 := (?, 

 and therefore, as both lenfes are to be alike, we have 9.12 

 X 2 = 1S.24 = F for each feparate focus; therefore 



18.24 



= 15.93 = Rofeach, and 15.93 '^ '-34= 21.35 

 K k 2 = r ; 



