96 OF PRESSURE AND EQUILIBRIUM. 



lar, the equation —^^fuxHx.^^dz, (Phil. Tr. 1819) 



y ^ dw 



the series, which it aflbrds, losiDg its convergence when x 



becomes large: here we find —zzif;, putting fyx^dxzzwx^; 



jjg _ dw _^ S/F~d>r _ _^iv d^z_d2/__2dw 2w _ 

 dx^ dx -^ ^3 ~^ V dr"3~"dr Idx xx ~~ 



— M?y— 2-H + — = — — (w;H Iv; and lastly -!-^ = 



■r ^^ j?a? o:^ >^ X ^ ^ '' dx"^ 



Ay 12w f 4 





Scholium 2. An important inversion of the Taylorian 

 theorem will be found at the end of this Book. 



248. Lemma F. Whenever one quantity 

 is dependent on another, their evanescent in- 

 crements are ultimately in a constant propor- 

 tion to each other. 



It is not sufficient to observe that, if y=:ax-\-hx"^ +cx" 

 -h dx^ + . . . the fluxion dy is=: djc (a + mhx "*~^ + ncx^-^ + 

 pdxP-^ + . . .) the quantity multiplying dx being constant 

 with regard to any small changes of the value of x and y ; 

 but it must also be shown, that the evanescent increment of 

 any quantity being supposed to be increased or diminished 

 in any given ratio, while it still remains evanescent, that 

 of another quantity depending on it will be increased or 

 diminished in the same ratio ; and this is not demonstrable 

 from the properties of the fluxions, strictly so called; but 

 it may be understood by observing that, whatever be the 

 form of the curve representing y by its ordinates, while 

 the absciss is x, a very small portion of it may always be 

 considered as approaching infinitely near to a straight line. 



