relation to the Figure of the Earth* 133 



p the cosine of the angle it makes with r. Then the potential of 

 the mass 



J J J 'VWTr^^cr^ 

 between proper limits, 



fi (* 2w f r /r 2 r 3 ' r i+2 \ 



By the conditions this is to equal zero for all values of c. Hence 

 it resolves itself into this, 



\ I I pr i + 2 V i d/jLdo)dr=0 



J-1J0 Je 



for all values of i in the series 1, 2, 3 . . . , P* being the ith La- 

 place's coefficient. The first term vanishes by the condition that 

 the total mass is zero. 



This last equation resolves itself into the condition that the 

 definite integral 



j: 



p r i+2 fa 



shall be a function of fi and co such that, when it is expanded 

 into a series of Laplace's functions, the ith function shall not 

 appear. Any other functions in the series not of the 2*th order 

 will of themselves vanish when multiplied by P* and integrated, 

 by a known property of Laplace's functions. 



4. By substituting for p in this last formula, integrating, and 

 putting r = a.w, we have, i being any number of the series 

 1, 2, 3 . . . ad infinitum, 



v ; W3 z + 4 2 + 5 / 



= some function which, when expanded into a series of Laplace's 

 functions, has no term of the t'th order. Such a function is the 

 following : — 



a«+ 8 (M + (I-1JMJ+ (i-2)M 9 + . . . + («-n)M n + . . .). 



Equating, then, these two series and extracting the (z + 3)rd root, 

 and leaving u alone on the left side, we have 



_ f M+fr , -l)M 1 + (i-2)M g +.. / 



U ~ 1 F G H 



L i + 3 + z + 4 + z+5 + '" 



When the right-hand side of this is expanded into a series of 



