694 Mr. stokes, on THE VARIATION OF GRAVITY 



Integrating by parts, and observing that "/ vanishes with ?', we get 



The last integral may be obtained by rationalization. If we assume R = tv - ^, and observe 

 that Qj = I5 Qi = cos\^, and that w = 1 when ^ vanishes, we shall find 



rtv-on, ,-/ , 1 '*" - cos \// , ,^ro-l , , w + 1 

 H -ZdZ = cos \l/ . log '- - (1 + cosur) - 2 cos xL ■ log . 



;„^ 



and 



When ^=1 we have Z = (2 - 2 cos \|/)" ' - (1 + cos \//), to = 1 + 2 sin — , ai 



fo^t-'Zd^ = - 2 sin I (1 - sin |) - cos x/, log fsin |(^1 + sin |] | . 



Putting /(\|/) for the value of y when ^ = 1, we have 



f(\^) = cosec — + 1 — 6 sin— — 5 cos \|/^ - 3 cos \|/ log |sin ^ 1+ sin - - I > (iS) 



In the expression for A?<, we may suppose the line from which 9' is measured to be the radius 

 vector of the station considered. We thus get, on replacing F{9', <p') by G~^i\g, and employing 

 the notation of Art. 22, 



^''=T^i:i''^s-M)''''^H'ix (*6) 



47rCr 



32. Let l^g = g + A'g'. Then A'g' is the excess of observed gravity reduced to the level of 

 the sea by Dr. Young's rule over calculated gravity ; and of the two parts g' and A'^ of which 

 A^ consists, the former is liable to vary irregularly and abruptly from one place to another, the 

 latter varies gradually. Hence, for the sake of interpolating between the observations taken at 

 different stations, it will be proper to separate Ag" into these two parts, or, which comes to the 

 same, to separate the whole integral into two parts, involving g' and A'g' respectively, so as to get 

 the part of Atf which is due to g' by our knowledge of the height of the land and the depth of 

 the sea, and the part which depends on i^'g by the result of pendulum-experiments. It may be 

 observed that a constant error, or a slowly varying error, in the height of the land would be of no 

 consequence, because it would enter with opposite signs into g' and i\'g. 



It appears, then, that the results of pendulum-experiments furnish sufficient data for the 

 determination of the variable part of the radius vector of the earth's surface, and consequently for 

 the determination of the particular value which is to be employed at any observatory in correcting 

 for the lunar parallax, subject however to a constant error depending on an error in the assumed 

 value of a. 



33. The expression for g'" in Art. 27 might be reduced to quadratures by the method of 

 Art. 31, but in this case the integration with respect to Y could not be performed in finite terms, 

 and it would be necessary in the first instance to tabulate, once for all, an integral of the form 

 Jo'/(C' '^"^V') ^K for 'values of \\r, which need not be numerous, from to ir. This table being 

 made, the tabulated function would take the place of /(\|/) in (iG), and the rest of the process 

 would be of the same degree of difficulty as the quadratures expressed by the equations (31) 

 and (4(5). 



34. Suppose At* known approximately, either as to its general features, by means of the 

 leading terms of the series (43), or in more detail from the formula (46), applied in succession to a 

 great many points on the earth's surface. By interpolating between neighbouring places for which 

 Am has been calculated, find a number of points where A;* has one of the constant values 



