Permanent Forms of Mathematical Expressions. 1 1 9 



A Sketch of the Limitations which are enforced upon 

 the Mathematical Forms of the Expressions for 

 Physical Quantities in a Continuous Medium in 

 consequence of the necessity for their Permanence 

 of Form. By R. F. Gwyther, M.A. 



{Received February 5th, 1895). 



In all parts of Applied Mathematics we find the same 

 forms of expressions occurring, and I propose in this sketch 

 to show how this recurrence arises from the fact that the 

 expression for a force, velocity or stress, must retain the 

 properties of such quantities, however we may shift our 

 axes of reference. The neglect of any consideration of 

 this kind at one time led to erroneous physical views, since 

 this similarity in form was connected with the physical 

 quantities themselves, instead of being treated as a mere 

 similarity of the mathematical expressions for their variations. 

 The well known example is the ancient error of considering 

 electricity to be a fluid, because in the measurement of its 

 effects we obtain expressions which look like, and can be 

 spoken of as if they were, those obtained by the motion of a 

 fluid. 



There is a mass of material which might be used in 

 illustration of this subject, and I have to choose between 

 the two plans of bringing illustrations from all sources, 

 which might be the more interesting, and of selecting them 

 from one subject so as to form a more continuous chain. As 

 the latter seems the more instructive I have chosen 

 it, and confine myself to questions of displacements and 

 stresses in a continuous medium. 



As an introduction, I take a case in which we deal with 



