occurring m Integrals of Partial Differential Equations. 297 



ducts to partial dliFerential equations, that is, to equations be- 

 tween three or more variables? and theory recognises at once 

 that these motions are not continuous, — that the path of a 

 moveable point is not necessarily given by a determinate 

 equation. 



Little or nothing has been done in this department of 

 science by the mathematicians of our own country, who do 

 not seem to have considered, that we cannot hope to reduce 

 to laws the multitude of facts that observation has accumu- 

 lated, and complete our knowledge of them, without cultiva- 

 ting the branch of calculation which corresponds to the nature 

 of the facts. The French academicians have shown that they 

 are aware of the importance of the subject, by the assiduity 

 with which they have of late attended to it. But perhaps the 

 information we have hitherto derived from their labours, is 

 hardly commensurate with the skilful and recJiercM modes of 

 investigation they have made use of. And this is observable 

 as well in those instances in which the integrations have been 

 completely effected, as in those in which they have not been, 

 obtained under a finite form. It has appeared to me that in 

 the former instances an important link in the chain of reason- 

 ing has been left out, and its place inadequately supplied by 

 the invention of a species of functions (discontinuous), which, 

 as pure analysis does not recognise them, cannot teach us any 

 thing. This omission, I think, is to be supplied by the deter- 

 mination of the form of the arbitrary functions, -prior to the 

 application of the integrals which contain them to any specific 

 case, and while the origins of the co-ordinates and of the time 

 yet remain indeterminate. When the question is about the 

 motion of the parts of fluids or solids, this form of the func- 

 tions points out the mode of action of the parts on each 

 other, which must be an action of a determinate character, 

 and is therefore to be ascertained by a determinate form of 

 the functions. For instance, in the discussion of the equations 

 for incompressible fluids, contained in the Phil. JSlag. and 

 Annals for August last, a specific form of the arbitrary 

 functions presented itself at once by the manner of perform- 

 ing the integration, and indicated a mode of action of the 

 parts of the ihiid on each other, which it was possible to verity 

 by referring to a very simple matter of fact. It was shown at 

 ihe same time how to arrive at the form of the arbitrary func- 

 tion, in an instance in which it did not immediately present it' 

 self. I proceed to employ the same method, to find the form 

 of the arliitrary functions in the integral which determines the 

 small motions of an elastic fluid, in which the pressure varies 

 as the density. 



Ar..V. Vol. G. No. 34.. Oct. 1829. 2 Q For 



