On Refraction. 395 



preceding. If, however, we compute one term only by this 

 approximation, and suppose the remainder of the series to be- 

 come geometrical, there will be a compensation of errors, which 

 for ordinary purposes, will be amply sufficient, and which may 

 often save us great labour in the computation of new terms. 



'ITius we have 1^ = 37.2, and i?H = 15; and 37.2 : 15 

 12ol 82 



:: 15 : 6, so that we may safely assume 6 as the quotient of a 

 geometrical progression nearly equivalent to the proposed series. 

 Its sum being equal to the last given term divided by 6 — 1, or 

 here to .000000016, making the whole .00004711, which must 

 undeniably be true to the last place, as far as the convergency 

 of the series is concerned. From this value of p A y we have 

 A y = .16031, y = .83969, hly = - .174842, and a; = 1 

 - ^ = 1 + .174842 X .00125254 - 1.0002190 ; but since 

 w = 1- A u= I -pAy, = .99995289, we have, for the new 

 value of s, ^ = .9997339, and v = .02307 ; and proceeding 



with tliese values to compute p A y, on the supposition A r = 

 .005, we have the series .00011535 + .00004079 + .00000697 

 + .00000260 + .00000071 + .00000028 + [.00000040] 

 = .00016710, instead of 00029388- .00004711 = .00024677, 

 consequently .005 is much too little, and we must multiply the 



terms by the powers of 1^ or •-^, and the sums will become 



.005 .005 

 .0002188 and .0002802 respectively, giving by interpolation 

 .006450 for r ; but it will be necessary to repeat the operation 

 with r=.0065, wiiich gives us .0002453, and requires the further 

 correction of .000025 only, making the whole refraction .011525 

 or 39' 37 ", which can scarcely differ above a second or two from 

 the truth, the terms actually computed showing that it jnust be 

 /ess than 39'45", and the remainder being capable of a very suf- 

 ficient estimation. Laplace's result is 39' 54",6 ; so that there 

 must probably be some numerical error in one of the computa- 

 tions ; at any rate the difference does not arise from the want of 

 convergence of the series. 



it may therefore be l.ft with confidence to the decision of 

 2 D2 



