THE EQUILIBRIUM OF FLUIDS. 139 



For this purpose we will substitute in the place of the co- 

 ordinates x' 9 y, z their values 



x = r cos 6' : y r sin & cos ix : z'=r sin & sin <&' ; 



and afterwards expand the function f(x' t y ', z) by Laplace's 

 simple method (Mec. Gel. Liv. iii. No. 16). Thus, 



f(x, y', z') =/"" +/" +/* + &c ....... +/< ......... (7) ; 



5 being the degree of the function f(x, y , z). 



It is likewise easy to perceive that any term f w of this 

 expansion may be again developed thus, 



/' =/.' r' +/,' r w +/; r" M + &c.; 



and as every coefficient of the last developement is of the form 

 U (i \ (Mec. Cel. Liv. iii.), it is easy to see that the general term 

 f (i) r i+2t may always be reduced to a rational and entire function 

 of the original co-ordinates x, y ', z'. 



If now we can obtain the part of V due to the term 



we shall immediately have the value of V by summing all the 

 parts corresponding to the various values of which i and t are 

 susceptible. But from what has before been proved (Art. 3), 

 the part of V now under consideration must necessarily be of the 

 form Y (i) ' } representing, therefore, this part by V t (i) j we shall 

 readily get 



[ 



J 



sn 



Moreover from what has been shown in the same article, it 

 is easy to see that we have generally 



-^r being the characteristic of any function whatever, and Y (i} 

 what Y' (i) becomes by substituting 0, to- the polar co-ordinates of 

 p in the place of 0', -or', the analogous co-ordinates of the element 



