AND THE FIGURE OF THE EARTH. 97 



and it is evident, by inspection of these functions, that that portion of the sum of their 

 partial differential coefficients, which arises from differentiating with respect to r 

 under the signs of integration, is equal to zero. This being the case, it is not neces- 

 sary to know the forms of the functions M a, N a, P a, &c., nor their numerical coeffi- 

 cients; but only the functions of ^u/, by which they are respectively multiplied ; and 

 these are Laplace's coefficients. 



Thus the equation of equilibrium at the surface would be 



Mo l/„ l\Na.l/, 6„.3\Pa ( ^ 15,. 5 „ 5\Qa 

 ^— 3r 



l/„ l\Na,l/. 6„,3\Pa /« 15., 5 „ 5\ 



4- g- (1 — y?) r^ m G, where — = — (1 -\- s fJi*'^ -{• A (ju^ -\- D ^^), suppose. 



Expand r, recollecting that N a, P a, and Q a are of the 1st, 2nd, and 3rd orders 

 respectively, and we have three equations to determine g, A and D. 



By differentiating C with respect to r, and eliminating N a, P a, and Q a by these 

 three equations, we have the resolved force in r, which divided by the cosine of the 

 angle of the vertical gives g exactly as in (23.). 



It is evident that this may be carried on indefinitely ; and to any order, without 

 finding g for the next lower order. 



MDCCCCXLI. 



