PEIRCE. — DERIVATIVE OF A SCALAR POINT FUNCTION. 341 



equal to V ; but at the point of tangency, the derivative of the function 

 U in any direction in the tangent plane of the V surface is zero, that 

 is, the normals to the U and V surfaces coincide, so that 



(7) 



and these familiar equations usually furnish some general information 

 about the problem independent of the value of V . As an extremely 



Figure 1. 



simple example we may take the familiar problem concerning the rela- 

 tive dimensions of an open tank of square base (x X x) and height y, 

 which shall hold a given quantity ( V=x 2 .y) of water and have the small- 

 est wet surface (U=x 2 + kxy). Here we have the curve D of the V 

 family, which has the given parameter, V , and are required to find that 

 member of the P, Q, R, S family which touches D. The equation (7) 

 becomes in this case 2y = x, and it appears (Figure 1) that the curves 

 of the two families which pass through any point of the line OM are at 

 that point tangent to each other. 



It is sometimes necessary to differentiate a point function, u, at a 

 point P, in the direction of the line through the point, along which 



