AT THE SURFACE OF THE EARTH. 693 



possibility of destroying the former by making allowance for the latter, let us take the earth such 

 as we find it, and consider further the connexion between the variations of gravity and the 

 irregularities of the surface of equilibrium which constitutes the sea-level. 



Equation (12) gives the variation of gravity if the form of the surface be known, and conversely, 

 (8) gives the form of the surface if the variation of gravity be known. Suppose the variation of 

 gravity known by means of pendulum-experiments performed at a great many stations scattered 

 over the surface of the earth ; and let it be required from the result of the observations to deduce 

 the form of the surface. According to what has been already remarked, a series of Laplace's coefBcients 

 would most likely be practically useless for this purpose, unless we are content with merely the 

 leading terms in the expression for the radius vector ; and the leading character of those terms 

 depends, not necessarily upon their magnitude, but only on the wide extent of the inequalities 

 which they represent. We must endeavour therefore to reduce the determination of the radius 

 vector to quadratures. 



For the sake of having to deal with small terras, let g be represented, as well as may be, by 

 the formula which applies to an oblate spheroid, and let the variable term in the radius vector be 

 calculated by Clairaut's Theorem. Let g^ be calculated gravity, r^ the calculated radius vector, 

 and put g=gc+ I^g, r = r^+ aH7i. Suppose Ag" and £\u expanded in series of Laplace's 

 coefficients. It follows from (12) that Ag' will have no term of the order 1 ; indeed, if this were not 

 the case, it might be shewn that the mutual forces of attraction of the earth's particles would have a 

 resultant. Moreover the constant term in Ag" may be got rid of by using a different value of G. 

 No constant term need be taken in the expansion of A^<, because such a term might begot rid of 

 by using a different value of a, and a of course cannot be determined by pendulum-experiments. 

 The term of the first order will disappear if r be measured from the common centre of gravity of 

 the mass and volume. The remaining terms in the expansion of A?< will be determined from those 

 in the expansion of Ag' by means of equations (8) and (l2). 



Let Ag-= G{v., + 1)3 -i-t), -1- ...), (42) 



and we shall have 



/\u = v.. + \v.s + \Vi + (-13) 



Suppose Ag* = GF{d, <p). Let \// be the angle between the directions determined by the angular 

 co-ordinates 6, (p and 9', (p'. Let (1 - 2^cos\^ -)- ^^)'' be denoted by R, and let Q,- be the coefficient 

 of t' in the expansion of R~' in a series according to ascending powers of ^. Then 



~' ^ ^-f:fo" ne', </)')«; sin e'de'd(j>', 



■Itt 

 and therefore if t be supposed to be less than I, and to become 1 in the limit, we shall have 



IttAm = limit otl'l'Tiff, (p')(5(Q, + ^ ^'Q,... + ^' ^ ^ ^'-'Qj + ...)sin e'dO'dfj)'. ... (44) 



I - 1 



Now assume 



and we shall have 



7=5?Q^+'rQ3-+'^r'-Q. 



^=5Q,+7tQ3...+(2i+i)r'-'«.+-; 

 x^^r ^d- r* =r3Q^ +?'«:. ••.+r^*«. +•••= rH«-' -(i- t^ih 



whence wc get, jjulting Z for R-' - Q„- (Qi, y = iif'('id .([^Z. 

 Vol. Vin. Pakt V. 4U 



