352 PROCEEDINGS OF THE AMERICAN ACADEMY. 



we may write the results of differentiating all the equations of (31) 

 with respect to x , %, z , in the form 



so that 



dx__L dy__M dz__N 

 dx ~ Q' dx ~ Q' dx ~ Q' 



L dp M dp N dp 



6p _ R'dxR'dyR'dz 

 dx L dx M dx N dx ' 

 B'dx~ + Ii'dy + B'dz~ 



(34) 



(35) 



and this is evidently equal to (9), the ratio of the directional deriva- 

 tives of p and ^o taken in the direction (s) in the (x, y, z) space in which 

 both i/ and z are constant. If (s, p), (s, x) represent the angles between 

 s and the directions of the gradient vectors of p and x respectively, 



dp __ k p • cos (s, p) 

 dx hxo ■ cos (s, x ) ' 



It is convenient, therefore, to define the derivative of a scalar point 

 function, u, with respect to another scalar point function, v, at any 

 given point in any direction (s), as the ratio of the directional deriva- 

 tives of u and v taken at the point in the direction s. 



Derivatives of this kind which frequently appear in two dimensional 

 problems in Thermodynamics and in Hydrokinematics, usually involve, 

 as has been said, a transformation from one set of coordinates to an- 

 other which is not orthogonal. 



Jeffehson Physical Laboratory, 

 Cambridge, Mass. 

 December, 1909. 



