Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 31 



27r' 



X<J>{t).W, 



. /n-2 \ 

 sm (— .) 



the equation immediately preceding will become 



^(0= ^I-:£ X {(j) (0) . f7o"' + 0(l) . t7;(W(^ (2) C7,». r' + &cc.\ 



[n — 2 \ 



which compared with the foregoing value of F^'\ will give by suppressing 



the factor '— , common to both, and equating separately the 



sm(-^.) 



coefficients of the different powers of the indeterminate quantity r the 

 following system of equations 



&c=...&c &c &c. 



for determining the unknown functions fo'-^, /<*', f/\ &c. by means of the 

 known ones f7"o*'\ Ui'-'\ ZJg*", &c. In fact the last equation of the system 

 gives U^^=fP, and then by ascending successively from the bottom to 

 the top equation, we shall get the values of fs^\ /,%, f}%, &c. with 

 very little trouble. It will however be simpler still to remark, that the 

 general type of all our equations is 



where the symbols of operation have been separated from those of 

 quantity and e employed in its usual acceptation, so that 



