REFRACTION. 



The refraftion deduced from Bradley's very neat and 

 finiple formula was, in a few years, adopted by nearly all 

 the aftronomers of enunence tln-ongliout Europe. Tlie ex- 

 treme facility with- which it nii^lit be computed, and the 

 correftioris applied, whether from the formula illtlf, or from 

 tables ready calculated for that purpofe, was a powerful 

 recommendation in its favour ; but its near agreement with 

 obfervation foon cllablifhed it. 



We muft now, without entering farther into detail of 

 minor improvements, proceed at once to the chapter given 

 by La Place on this fubjeft, in his " Mecanique Celefte," 

 vol. iv. p. 231, where he has drawn, from an inveligation 

 which we cannot undertake to exhibit In this place, the fol- 

 lowing general formula, for expreiling the refraftion for all 

 angles of elevation above I2 decimal, or 10.8 fexagefimal 

 degrees, viz. 



a p 



tan. z 



+ 



0.76(1 +/. 0.00375) 



ap tan. 2 



' ^ . 0.0012C2C.' ■ — 



0.76 



(1 + 2 cof.-z) tan. 3 



fi 



cof.= 



[0.76(1 + /.O.O0375)]- 



where all the quantities are known except r and a ; wliich 

 latter rcprefents a conftant co-efficient ; w'z. z is the ob- 

 ferved zenith diftance under the barometric preffure p, in 

 metres, and t the temperature of the centigrade thermo- 

 meter, r being the refraftion : all, therefore, that is re- 

 quired for determining r, is the value of the unknown co- 

 efficient a ; which is to be drawn from obfervations on the 

 circumpolar ftars, in the following manner. 



Let Z be the dillance of one of tliefe ftars from the 

 zenith, in its fuperior meridian paflage ; Z', this diftance at 

 the inferior meridian palfage, obferved from the fame point 

 of the terreltrial furface ; r and r', the correfponding re- 

 fraftions. Now all the other quantities, except a, being 

 knovs'n, we may, for the fake of fimplicity, put the above 

 formula for both paffages under this form, 



r = Aa -f Ba', ;■' = h!'a -f- B'a' ; 



where A, B, A', B', are all known quantities. Writing 

 aUo Z, Z', for the obferved zenith diilances, the true zenith 

 diftance, D, correfted for refraftion, will be 



Z -f A« -t- B(z% and Z' + A's + Y>'a, 



which are, therefore, now equal to each other ; confequently, 

 by addition, we have 



Z + Z' -I- (A + A') « + (B + B') a-=2V>; 



in which all the quantities are known, except a and D. 

 But by repeating fimilar obfervations on fome other ftar, 

 and denoting by Z", Z'", A", A'", B", and B"', the fimilar 

 quantities before reprefentcd by Z, Z', A, A', &:c. alfo a 

 and D remaining the fame for all ftars obferved in the fame 

 place, we lb all have thtfe two equations ; 



Z + Z' + (A + A') a + (B + B') a' = 2 D, 



Z" + Z'" -h (A" + A'") a + (B" H- B'") a-^lV); 



from which it is obvious, the conftant co-efficient a may be 

 obtained by the ufu.il methods of elimination. 



In the above operation, however, we have fuppofed the 

 polar diilances of the fame ftar to be the fame for its fu- 

 penor and inferior paffiige; whereas we know that, in con- 

 sequence of the effect of preceffion, nutation, and aberra- 

 tion, this diftance is conftantly varying ; and >ve ought, 

 therefore, to introduce thefe variations into the above equa- 

 tion. But our objeft being merely to give a general view 

 of the principles made ufe of for the determination of the 



co-efficient a, we liave not thought it neccfTary to enter fo 

 Itnttly into the mnuitia of the computation. It appea-s 

 from the above, that the conftant co-efficient a may be dc. 

 termined by means of four obfervations on two different 

 circumpolar ftars; and confequently, that every fuch fet of 

 obfervations ought to produce the f.me refult, or the fame 

 value of a. Confidering, however, the extreme accuracy 

 required in fuch cafes, both in the inftruments and the ap- 

 plication of them, fome little difagreement is to be expeftcd ■ 

 .and indeed one is furprifcd to fee it fo fiv.all, as it has beer! 

 found to be in various obfervations undertaken for this pur- 

 pofe, and the mean of which we have every reafon to coii- 

 /ider as perfeftly corred ; and which is ftated by M. Biot, 

 who has interefted himfelf very much on this fubjed, at 

 iS7".24 for the decimal divilion, or 6o''.666 for the fexa- 

 gefimal. 



But now, in order to Amplify our firft formula, by taking 

 / = 0.76 metres, and / = o, this may be put under the 

 form. 



r = fl tan. Z (i 

 tan. Z ; 



0.00125254 

 cof.'K 



') 



-t- i«'fin. 1" 



I -f- 2 Cof.' Z 



cof.^ z 



in which, fubftituting for cof.' z, its value ^ , and 



I -h tan.^a 

 the proper numerical value of a, as above found, as alfo of 

 lin. i", the whole is reduced to the following form, wz. 



r= 0.99918761 . atan. Z — O.ooi 105823 . a tan.^Z ; 

 which latter form M. Biot has fhewn to be equivalent to 

 /•= l87".24.tan. (Z - 3.25 r) for the decimal divifion; 

 r = 6o".666 . t.an. (Z - 3.25r) for the fexagefimal diviiion. 

 But the reduftion of it to this form would occupy more 

 fpace tlian can be allowed for this article. This laft form 

 IS as fimple as can be defired, from which the following rule 

 in words may be deduced, -uhz.. The vefradion under the fame 

 barometric prejfure, and the fame degree of temperature, is pro- 

 portional to the tangent of the apparent zenith dijiance of the 

 ftar, dinuni/Ioed by -3,^ times the refraHion. 



It muft be remarked, however, that the formula r = A 

 tan. (Z - 3.25/-), though it exhibits the law of refradion 

 in as fimple a form as can be defired, is not well adapted 

 for calculation, in confequence of /- entering on both fides 

 of the equation ; and aftronomers have, therefore, o-iven 

 dift'erent methods of rendering the above formula more 

 commodious. In order to which, it is firft put under the 

 form., 



tan. n r = tan,- n R tan. (Z — nr)\ 



R reprefcnting the refraaion, anfwering to Z = 90°. Let 

 us now add fucceffively to both fides of this equation, the 

 quantities -f tan. n r tan.^ n R - tan. « r tan.^ n R, and we 

 ihall have, 



tan. nr (I + tan.-nR) = tan.' n R [tan. (Z-n»-)-f tan. nr], 

 tan. « A- (I — tan.-- n R) = tan.' n R [tan. (Z -nr)- tan. nr]. 



Now, dividing thefe equations, member by member, n r will 

 be ehminated, and we obtain 



I -h tan.'^/z R ^ tan. (Z — nr) -f tan.nz- 

 l-tan.'nR "" tan, (Z —nr)— tan. nr' 

 I fin. Z 



cof. 2 ?! R ~ 

 whence we draw 



fin. {"L — znr) 



fin.(Z-2nR)' 

 = cof. 2 « R . fin. Z. 



Now 



