690 Mr. stokes, on THE VARIATION OF GRAVITY 



27. Let it be required to reduce gravity g to the gravity which would be observed, in the 

 altered state of the surface, along what would then be a surface of equilibrium. Let the correction 

 be denoted by g — Sg'", where g' is the same as before. The correction due to the alteration of the 

 coating in the manner considered in Art. 20 has been shewn to be equal to 



2 7r^-67r2-r-i-, 

 2l + 1 



and the required correction will evidently be obtained by replacing S by ax. Putting for x^ its 

 value got from (35) we have 



® ® {Hi + I) p - Sa ' I {2i + l)p-3a] 



which gives, since 27rcr2^j = SttctA =g', and G = ^wpa, 



Sg"'= G— S , ■ ^ , — - (36) 



^ 2p (2t + 1) p - 3(T a 



If we put (T = 2I, p = 5^, a = 4000, and suppose h expressed in miles, we get 



Sg" = G. 2 -^ -'— = G X .00017 (- 4.5^0 + ^1 + -^S/t. + .29OA3 + .214./ti + ...). ...(37) 



^ 8S000 11 j- 2 



Had we treated the approximate correction Sg" in the same manner we should have had 



3g" = G — ^2 — = G X .00017 (3^0 + ''1 + -6*2 + .4.29A3 + .333ki + ...) 



Zpa 2j + 1 



whereas, since A: = 3 ( 1 ) > we get 



kg" = G — S /~ '^' ■ '' = G X .00017(1.636/(0 + .545^1 + .327A2 + .234^3 + .182^1 + ...).. .(38) 

 2|ua (2j + \) p 



The general expressions for 3g"', Sg'', and kg" shew that the approximate correction kg" agrees 

 with the true correction 3g"' so far as regards terms of a high order, whereas the leading terms, 

 beginning with the first variable term, are decidedly too small ; so that, as far as regards these 

 terms, 3g"' is better represented by Sg" than by kg" . This agrees with what has been already 

 remarked in Art. 25. 



If we put g-- §•'+ 3g-"'=g'_ , and suppose G and e determined by means of g^^^, small corrections 

 similar to those already investigated will have to be applied in consequence of the omission of the 

 quantity g' — 3g"' in the value of g. The correction to e would probably be insensible for the 

 reason mentioned in Art. 18. If we are considering only the variation of gravity, we may of course 

 leave out the term /i,,. 



The series (37) would probably be too slowly convergent to be of much use. A more 

 convergent series may be obtained by subtracting kg" from Sg", since the terms of a high order in 

 Sg"' are ultimately equal to those in kg". We thus get 



Sg-'" = kg" + G X .00017 (-6.I36A0 + .455^1 + .123A, + .O56A3 + .032^4 + ...) {SO;) 



which gives g'" if g" be known by quadratures for the station considered. 



Although for facility of calculation it has been svipposed that the sea was first replaced by a 

 stratum of rock or earth of less thickness, and then that the elevated portions of the earth's 

 surface became fluid and ran down, it may be readily seen that it would come to the same thing if 

 we supposed the water to remain as it is, and the land to become fluid and run down, so as to 

 form for the bottom of the sea a surface of equilibrium. The gravity g^^^ would apply to the 

 earth so altered. 



