Astronomical and Nautical Collections. 143 



+ . 6517?-' 5 -+■ . . . Now the value of /•' cannot be very accu- 

 rately obtained from these coefficients, without a liberal em- 

 ployment of the method of logarithmic differences, finding the 

 results derived by it from the first three, the middle three, and 

 the last three terms, and comparing these with each other; and 

 in this manner it seems natural to suppose that we might easily 



come within about . J_ of the truth. (Coll. VIII.) The best in- 

 500 



ference of this kind, however, that has been obtained, was 



r =40' 15", which is too much by about . 



J 130 



If still greater accuracy were required, we might compute a 

 greater number of the coefficients of the series, or we might 

 separate the computation into two or more parts : but it would 

 be a little troublesome to adapt the new values of Z, and its de- 

 rivatives, either to the diminished magnitude of the density z, 

 or to a value of p, diminished in the same proportion ; so that if 

 the actual density at the time in question were called unity, the 

 refractive density might still be truly represented by 1 + p ; 

 observing also to make the remaining portion A z also equal to 

 unity : and in this case the values of Z, and of its powers, only 

 would require to be changed in the subsequent computation. 

 This operation has been somewhat more negligently performed 

 in the Astronomical Collections ; but its object then was merely 

 to show the convergence of the series, and that object was 

 obtained. 



b. With the values m =r 766, p — , and =4.621, 



r 3540 mp 



we obtain Z = 3.621, Y = i^i, Z' - YZ = 17. 190, 



P 



Z»=±YZ°- + Y*Z = 452 - 916 , Z>" = ll^i, and *£ 

 p p« p y 



3714095 ,W A t - i Z , , Z' .„ " 



; and for the equation 1=. — r + — pr • + . . . , 



p* n 2 24 



we have 1 = 1.8105/ + .7162r'2 + .6290/3 + .7760/* + 



1 . 0231/ 5 + . . . ; and if we make / = 44, we shall have the 



