48 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



dp are referred to the displacement along an element of line ds. Forming the sum 

 for any succession of line elements, we get the equations referred to a curve of finite 

 length, namely, from (c), section 34, 



(a) <f> 2 - & = - I adp 



and from (c), section 37, 



(*) 



X*2 

 ^1 



The first of these equations gives the difference of potential, i. e., the difference 

 of dynamic height, between the isobaric surfaces of pressures p 2 and p x . The 

 second gives the difference of pressure between two equipotential surfaces of 

 potentials < x and <f> 2 . 



The dynamic sense of the integrals forming the second member of equations 

 (a) and (b) is easily found, as we have 



acip = a Gds pd<f> = pgds 



Thus the integral in (a) is the line-integral of the force of pressure per unit-mass. 

 The integral in equation (b) is the line-integral of the iorce of gravity per unit- 

 volume. On the other hand, the differences appearing on the left side of the 

 equations $ 2 4>i ar >d pi p\ are the line-integrals of the force of gravity per unit- 

 mass and of the force of pressure per unit-volume. Equation (a) thus shows that 

 the force of gravity and the force of pressure, both referred to unit-mass, have 

 oppositely equal line-integrals. In the same way, equation (b) shows that the force 

 of gravity and the force of pressure, both referred to unit-volume, have oppositely 

 equal line-integrals. One of the two oppositely equal line-integrals can always be 

 expressed in finite form, namely, that of the force of gravity per unit-mass and that 

 of the force of pressure per unit-volume. 



The equations (a) and (b) enable us at once to derive a fundamental property 

 of the integrals appearing on the right side. The values <f>y and <f> 2 of the potential 

 in the end-points 1 and 2 of the curve depend only upon the situation of these points 

 1 and 2, and not upon the course of the curve s joining them. The same is the 

 case with the values p x and p 2 of the pressure in the same two points. The integrals 

 on the right side, therefore, must have the same property. Hence we conclude: 



Under statical conditions the line-integral of the force of pressure per 

 unit-mass 



( C ) - r~adp 



l/p, 



as well as the line-integral of the force of gravity per unit-volume 



X*2 

 .1 



are independent of the course of the curve and dependent only upon the posi- 

 tions of its end-points. 



As a corollary we get this other theorem: 



Under statical conditions the line-integrals of the force of pressure per 

 unit-mass (c), as well as the line-integral of the force of gravity per unit-volume 

 (d) are zero for every closed curve. 



