Mr. stokes, on THE VARIATION OF GRAVITY 



14. Before we consider how the numerical value of the earth's ellipticity is to be determined, 

 it is absolutely necessary that we define what we mean by ellipticity ; for, when the irregularities of 

 the surface are taken into account, the term must be to a certain extent conventional. 



Now the attraction of the earth on an external body, such as the moon, is determined by the 

 function V, which is given by (10). In this equation, the term containing r"^ will disappear if r 

 be measured from the centre of gravity; the term containing r"*, and the succeeding terms, will 

 be insensible in the case of the moon, or a more distant body. The only terms, therefore, after 

 the first, which need be considered, are those which contain r"". Now the most general value of m^ 

 contains five terms, multiplied by as many arbitrary constants, and of these terms one is -^ - cos' 9, 

 and the others contain as a factor the sine or cosine of i^ or of 2 (p. The terms containing sin (p or 

 cos d, will disappear for the reason mentioned in Art. 12 ; but even if they did not disappear their 

 effect would be wholly insensible, inasmuch as the corresponding forces go through their period in 

 a day, a lunar day if the moon be the body considered. These terms therefore, even if they ex- 

 isted, need not be considered ; and for the same reason the terms containing sin 2(p or cos 2(p may 

 be neglected; so that nothing remains but a term which unites with the last term in equation (10). 

 Let 6 be the coefficient of the term ^ - cos- 6 in the expansion of u : then e is the constant which 

 determines the effect of the earth's oblateness on the motion of the moon, and which enters into 

 the expression for the moment of the attractions of the sun and moon on the earth ; and in the 

 particular case in which the earth's surface is an oblate spheroid, having its axis coincident with 

 the axis of rotation, e is the ellipticity. Hence the constant e seems of sufficient dignity to deserve 

 a name, and it may be called in any case the ellipticity. 



Let r be the radius vector of the earth's surface, regarded as coincident with the level of the 

 sea; and take for shortness m {f{0,(p)\ to denote the mean value of the function/ (6,<p) 



throughout all angular space, or — [„" f^'^"/ (fi,(j)) sin 9 dOdcp. Then it follows from the theory 



of Laplace's coefficients that 



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e = — m {(1- sin'Z)r|, (26) 



4 a 



I being the latitude, or the compliment of 9. To obtain this equation it is sufficient to multiply 



both sides of (8) by (1 - cos^ 9) sin 9d9d(p, and to integrate from 6 = to = tt, and from 



^ = to (^ = 2 TT. Since ^ - cos^ is a Laplace's coefficient of the second order, none of the 

 terms at the second side of (S) will furnish any result except Mo, and even in the case of n^ the 

 terms involving the sine or cosine of (h or of 2d) will disappear. 



15. Let g be gravity reduced to the level of the sea by taking account only of the height of 

 the station. Then this is the quantity to which equation (12) is applicable; and putting for u. its 

 value we get by means of the properties of Laplace's coefficients 



45 

 G = m(g),G{^m -e) = --m {(i-sin=/)^| (27) 



If we were possessed of the values of g at an immense number of stations scattered over 

 the surface of the whole earth, we might by combining the results of observation in the 

 manner indicated by equations (27) obtain the numerical values of G and e. We cannot, however, 

 obtain by observation the values of g at the surface of the sea, and the stations on land where 

 the observations have been made from which the results are to be obtained are not very numerous. 

 We must consider therefore in what way the variations of gravity due to merely local causes are 

 to be got rid of, when we know the causes of disturbance ; for otherwise a local irregularity, 

 which would be lost in the mean of an immense number of observations, would require undue 

 importance in the result. 



