OF PRESSURE AWD EQUILIBRIUM. 95 



equation fyxAszzsxzi^x. 7" (^dj2 ^jyi; y P"ttiags=ax + 



6x3 +cx= + . . . and g=^^^^=.(l +J,. + ...), and 



substituting for the powers of «, we obtain a value of y 

 which affords that of fi/xdx, and by comparing its terms 

 with those of the value of sx, we determine the successive 

 coefficients. (Suppl, Enc. Brit. Art. Cohesion). But the 

 series is often inconvenient for want of convergence: we 

 may therefore supply its defects by means of the Tayloriaa 

 theorem, taking the successive fluxions of s at the point 

 of the curve where we find it necessary to abandon 



the series: thus yord.r =:sdj:-\-j:ds, y^- + j-, j'^y »' 



.ds . d« s(\x ._, ^ ^ dd^ s y 2s 

 d-r-=dy — - + or, ifl— s2=m2,-_-_ Z _^ — 



, , , . 5ds , dw 5* *y d^^ 

 and au bemg zz , and -t" = — ^, -^ — iz 



M 



V 2s s^ s"y 3y 65 ,^ s d^s 



w 2f #3 ^2y Sy 6s ^ 



+ — ^+-e- . that IS, since 1 + ^2- 



1 y 2t t^ Sy 6s ,,.,.. 



— , -^r L + ;1r r» ^^^ the fourth fluxion 



1^2 ' ii3 jQ X XX x^ 



may be found in a similar manner, if its value be required : 

 but the first three will be fully sufiicient, provided that the 

 curve be divided into small parts, even though they may be 

 much larger than those which Laplace has employed in the 

 Connaissance des Tems for 1810: and this method will 

 probably be found at least as convenient as the much more 

 elaborate process of Mr. Ivory. (Suppl. Enc. Br. IV). We 

 may take, for another example of a difficulty precisely simi- 



