58 DEFINITE INTEGRALS. [INT. III. 



pendent of the path, SI must vanish, which can only happen for all 

 possible choices of SOD, By, Sz if 



_ = _ = _ = 



'by dz dz dx ~ dx dy ~ 



that is, if the curl of R vanishes everywhere. In case this con- 

 dition is satisfied, / depends only on the positions of the limiting 

 points A and #, and not on the path of integration. If A is given, 

 / is a point-function of its upper limit B, let us say c/>. If B is 

 displaced a distance s in a given direction to B', the change in the 

 function < is 



fa -<t> B =( B (Xdx + Ydy + Zdz), 



J B 



and the limit of the ratio of the change to the distance 



lim *g-fr = j* Z !jg + F ^ + z 



s=0 s ds as as as 



is the derivative of $ in the direction s. 



If we take s successively in the directions of the axes of co- 

 ordinates, 



<^_ Y ty-V fy-7 



- -^A- j t-\ "~~ -* * >"\ ^9 



as dy dz 



that is, R is the vector differential parameter of the scalar 

 function <f>. 



Accordingly the three equations of condition equivalent to 

 curl R = are simply the conditions that X, Y, Z may be repre- 

 sented as the derivatives of a point-function. In this case the 

 expression 



is called a perfect differential. 



From the definition of the parameter of a scalar point-function, 

 we see that the magnitude of the parameter is inversely propor- 

 tional to the normal distance between two infinitely near level 

 surfaces of the function. Such a pair of surfaces will be called a 

 thin level sheet or lamina. For this reason a vector point-function 

 that may be represented everywhere in a certain region as the 



