132 Permanent Forms of Mathematical Expressions. 



elasticity, to make the sketch of the subject more compact, 

 but illustrations may be found throughout the subjects of 

 Applied Mathematics, shewing that quantities subject to 

 permanency of form are limited in their form of mathematical 

 expression and by this means we can, not of course explain, 

 but, at least, place in one connection the various analogies 

 which shew themselves throughout the subject of applied 

 mathematics even in cases where the reason of the analogy 

 is far to seek. I hope, however, to have shown in what way 

 mathematical results may be independent of the physical 

 hypotheses which may have been employed to obtain them, 

 but it would require too much space to treat this question 

 fully, or to discuss its bearing upon the modern treatment 

 of the principles of mechanics. 



