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Report on the present State of the Analytical Theory of Hydro- 

 statics and Hydrodynamics. By the Rev. J. Challis, late 

 Fellow of Trinity College Cambridge. 



The problems relating to fluids, which have engaged the atten- 

 tion of mathematicians, may be classed under two heads, — those 

 which involve the consideration of the attractions of the con- 

 stituent molecules, and the repulsion of their caloric ; and those 

 in which these forces are not explicitly taken" account of. In 

 the latter class the reasoning is made to depend on some pro- 

 perty derived from observation. For instance, water is observed 

 to be very difficult of compression ; and this has led to the 

 assumption of absolute incompressibility^ as the basis of the 

 mathematical reasoning : air at rest, and under a given state of 

 temperature, is observed to maintain a certain relation between 

 the pressure and the density ; hence the fundamental property 

 of the fluid which is the subject of calcvilation is assumed to be 

 the constancy of this relation, to the exclusion of all the circum- 

 stances which may cause it to vary. The fluids treated of in 

 this kind of problems are rather hypothetical than real, yet not 

 so different from real fluids but that the mathematical deduc- 

 tions obtained respecting them admit of having the test of ex- 

 periment applied. I propose in this Report to confine myself 

 entirely to problems of the second class, — those in the common 

 theory of fluids. The reasons for making this limitation are, 

 that both kinds together would afford too ample matter for one 

 Report, and that those which I have selected are distinguished 

 from the others by the different purpose in regard to science 

 which correct solutions of them would answer : for the treat- 

 ment of any hydrostatical or hydrodynamical questions which 

 involve the consideration of molecular attraction and the repuli- 

 sion of heat, must proceed upon certain hypotheses respecting 

 the mode of action of these forses, and the interior constitution 

 of the fluid, as these are circumstances which from their nature 

 cannot be data of observation ; and hence, assuming the ma- 

 thematical reasoning founded on the hypotheses to be correct, 

 a satisfactory comparison of the theoretical deductions with 

 facts must serve principally to establish the truth of the hypo- 

 theses, and so to let us into secrets of nature which probably 

 could never be known by any other process. But when the 



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