hersey: derivatives of physical quantities 621 



face (1) from a knowledge of another, although the equation of 

 the surface is not available. The interest of the problem to the 

 physicist lies in the fact that he may wish to learn the value of 

 a derivative not readily accessible to experiment, in a case 

 where some other derivative of the same quantity can easily be 

 observed. It will be shown that a definite solution can always 

 be obtained, provided certain dimensionless products of the N 

 quantities are held constant. 



Other classes of relations among derivatives. The proposition, 

 that relations may be found connecting the derivatives of quan- 

 tities in the absence of a primitive equation, is not new. There 

 are two other classes of such relations. One consists of mathe- 

 matical identities, applicable to any set of related quantities, 

 whether physical or not. To this class belongs the identity 



d dQ d dQo 



as well as the triple product relation 



dQ DQt bQ, 



(3) 



dQi dQ 2 dQ c 



= - 1 (4) 



The other class comprises relations requiring the explicit use of 

 physical laws, such as the two laws of thermodynamics, or 

 Hamilton's principle. To this class belong Maxwell's four ther- 

 modynamic relations, and the reciprocal relations of generalized 

 dynamics. 2 The relations to be presented here are of a nature 

 intermediate between the other two classes, in that they require 

 a knowledge only of the dimensions of the quantities. 



Derivation of the new relation. The present result depends 

 upon and is a corollary to Buckingham's n-theorem, 3 according 

 to which any complete physical equation is reducible to the 

 form 



funct.(n 1 , n 2 , . . . TU) = (5) 



2 J. J. Thomson, Applications of dynamics to physics and chemistry, Chap. V. 



3 This Journal, 4: 347-353. 1914; Phys. Rev., 4: 345-376. 1914; Trans. Am. 

 Soc. Mech. Engs., 37: 263-296. 1915. Any one who can sufficiently visualize the 

 meaning of the n-theorem will be able to treat each concrete problem by itself, 

 dispensing with the formulas of the present paper save as a check. 



