ON THE MATHEMATICAL THEORY OF FLUIDS. 225 



Supplementary Report on the Mathematical Theory of Fluids, 

 By the Rev. J. Challis, Plumian Professor of Astronomy 

 in the University of Cambridge. 



The object of the first Report which 1 read before the Associ- 

 ation was to sketch out the processes of calculation, and exhibit 

 the results obtained in the applications of analysis to fluids which 

 were supposed either to be incompressible, and therefore of uni- 

 form density ; or compressible in such a manner that the density 

 is, under all circumstances, the same where the pressure is the 

 same. Such fluids do not exist in Nature. All liquids are com- 

 pressible in some degree, and the pressure in every aeriform fluid 

 varies as well with the temperature as with the density; yet these 

 hypothetical fluids are in a mathematical sense intimately allied 

 to existing fluids. The results which calculation gives on the 

 supposition of incompressibility admit of comparison with facts 

 observed in the equilibrium and motion of water ; and the laws 

 of pressure, motion, and propagation of motion, arrived at in the 

 mathematical treatment of the imaginary fluid, whose pressure is 

 conceived to depend on the density alone, are first approxima- 

 tions towards a knowledge of w^hat actually takes place in air. 

 The comparison of the calculated results with fact and experi- 

 ment, in these normal cases, serves to show the degree of influence 

 to be attributed to the modifications which the fundamental pro- 

 perties of the imaginary fluids must undergo, to make them agree 

 more nearly with those of real fluids. Of late years mathema- 

 ticians have introduced such modifications into their theories, 

 by reasoning from certain hypotheses, respecting the interior 

 constitution of bodies, and the mechanical action of their mole- 

 cules, for the purpose of treating mathematically of matter as it 

 exists in Nature, and tracing to causes beyond the reach of di- 

 rect observation and experiment the various sensible phaenomena 

 which it presents. I endeavoured in a second Report to give 

 some account of the general principle of such theories, and to 

 explain how they serve, by a satisfactory comparison of the the- 

 oretical results with experiments, to establish the truth of the 

 hypotheses on which the mathematical reasoning is based, and 

 so to make known, respecting the intimate constitution and un 

 seen conditions of bodies, something which could not be ascer- 

 tained by observation alone ; as, in an instance in some respects 

 analogous, mathematical calculations applied to electrical phae- 

 nomena are considered to prove the existence of fluids whose 

 VOL. V. — 1836. . Q 



