Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 17 



the series just given are expansions, that whatever number i may re- 

 present, Qi*** will be immediately deduced from Q*'> by changing 9, sr, 



9', -sr' into 0„ "sr,, 9/, ^r,'. But since the quantity - is indeterminate, 



and may be taken at will, we get, by equating the two values of . , 



. f 

 and comparing the like powers of the indeterminate quantity -, 



If now we multiply the equation (6) by the element of a spherical 

 surface whose radius is unity, and then by Q<*' = Q/*>, we shall have, 

 by integrating and extending the integration over the whole of this 

 spherical surface, 



fdf.dwQ"^ r® = fdfx, d-ar, Q/** { F/"' + Y^ + F*^' + &c. } . 



Which equation, by the known properties of the functions Q**' and Y^^\ 

 reduces itself to 



when h and i represent different whole numbers. But by means of a 

 formula given by Laplace {Mec. Cel. Liv. iii. No. 17.) we may imme- 

 diately effect the integration here indicated, and there wiU thus result 



"-2^ + 1-^^ ' 



F;'<*> being what Fi''" becomes by changing 9^, tsti into 0,', •ar/, and as 

 the values of these last co-ordinates, which belong to p, may be taken 

 arbitrarily like the first, we shall have generally F,**', except when 

 h = i. Hence, the expansion (6) reduces itself to a single term, and 

 becomes 



F® = F®. 



We thus see that the function F<'' continues of the same form even 

 when referred to any other system of axes X„ F„ Z„ having the same 

 origin O with the first. 



This being established, let us conceive a spherical surface whose center 

 is at the origin O of the co-ordinates and radius r', covered with fluid. 

 Vol. V. Part I. C 



