344 PROCEEDINGS OF THE AMERICAN ACADEMY. 



pressible in terms of the other : the gradient of v = f x s — 4 xy 2 is 

 equal at every point of the ay plane to the square of the gradient of 

 u = x % — if. If u and v are orthogonal functions of x and y, the 

 product of their gradients is equal to the Jacobian, 



du dv. du dv 

 dx dy dy dx 



The differential equation 



(sy+SHS)'-*. 



which leads to systems of parallel surfaces, is of standard form. Its 

 complete integral is 



u = ax + by + z VF — a 2 — b 2 + d, 



where a, b, d are arbitrary constants, and from this the general 

 integral may be obtained in the usual manner. 



If a direction s be determined at every point of a given region, T, 

 by some law, the derivative of the function u becomes itself a scalar 

 point function in T, and if this is differentiable, it may be differentiated 

 at any point in any direction, say s. It is usually convenient to 

 define s by means of three scalar point functions, /, m, n, the sum of 

 the squares of which is identically equal to unity, and which represent 

 the direction cosines of s. In this connection it is well to notice that 

 if s has the direction at P of the tangent of a continuous curve which 

 passes through the point, if P' be a point near P on the tangent and 

 P" a point near P on the curve, and if U is any differentiable scalar 

 point function, 



Up» - Up Up> - Up 



PP" ' PP' 



have the same limit, as P' and P" approach P, that which has been 

 defined as the derivative of U at P in the direction s. If, then, 

 du/ds is differentiable 



d_ 



(du\ _ d f du du du\ 



\ds J dx \ dx dy dzj 



_ , d' 2 u d % u d 2 u du dl du dm du dn 



dx 2 dx-dxi dx-dz dx dx dy dx dz dx' 



(11) 



