OF THE EQUILIBRIUM OF FLUIDS. 193 



tubes or branches belonging to it, the water will stand at 

 the same height in all. 



313. Lemma A. The partial variations' 

 ^y (Ku) and ^x {^yu) are equal. 



For, when the variation of u is taken with respect to a;, 

 the quantities depending on y remain unaltered, and the 

 process leads to the same result, when the variation is 

 afterwards taken with respect to y, as if it had been in- 

 verted. For example, if Mzzx'y, ^j^u — mx"'^-^ ^x.y^, and 

 8y {^^u)=zmnx"'-^ y^'-^^xhjzz^^ {^yu): again, i£u=ax--\- 

 bi/, Bxu:=i2ax^r, ^y cSaM)=0; ^yU=2hj^y, g^ (^yu)=0 : 

 and ifuzzx^'y^zP, the same results will be obtained, for the 

 variations with respect to x and y, as if z were a constant 

 quantity. 



314. Lemma B. If hi=mj; + M^i/-hUz, 

 we nave -5— ~t~» "1^-=-^— , and -§— =-?r-. 



by hx Ez gx ' dz Sy 



For ^^uzzMx, and SyU=Mhj, and dy{Ku)=^y (Mx)= 



g,:(M(313)=g^(M5^i/)zz^a:r5y=^ ^y^x; conse- 



quently -vr-^-r— : and m the same manner the other 



ox oy 



equations are obtained, by comparing the variations in 

 pairs. 



315. Corollary. An exact variation, 



containing two or more variable quantities, 



must always be conformable to the condition 



of this proposition. 



Scholium. This condition of integrability was first 

 laid down by Nicolas Bernoulli, in 1728.] 



