132 THIRD REPOtlT— 1833. 



basis of calculation, as in the questions that will come before 

 us, is some observed and acknowledged fact, solutions which 

 satisfy experiments will first of all serve to confirm the truth of 

 the mathematical reasoning, and then give us confidence in the 

 theoretical results, which, as often happens, cannot readily re- 

 ceive the test of experiment. Calculations of this kind do not 

 add much to our conviction that the facts applied as the test of 

 the theory are really consequences of those which are the basis 

 of it. For instance, we feel satisfied, independently of any ma- 

 thematical reasoning, that the motions of waves on the surface 

 of water are consequences of the incompressibihty of the fluid, 

 and the law of equal pressure. But the purpose which these 

 calculations answer of confirming methods of applying analysis 

 is very important, particularly in regard to the higher class of 

 physical questions, which M. Poisson has proposed to refer to 

 a distinct department of science, under the title of Math4ma- 

 tique Physique, viz. those that require in their theoretical treat- 

 ment some hypotheses respecting the interior constitution of 

 bodies, and the laws of corpuscular action : for in questions of 

 this nature, as well as in problems in the common theory of 

 fluids, the mathematical reasoning conducts to partial differen- 

 tial equations ; and if the method of treating these, and of 

 drawing inferences from their integrals, be established in one 

 kind, it may be a guide to the method to be adopted in the 

 other. It is plainly, then, desirable that the mathematical pro- 

 cesses be first confirmed in the cases in which the basis of rea- 

 soning is an observed fact, that the reasoning may proceed with 

 certainty in those cases where it is based on an hypothesis, the 

 truth of which it proposes to ascertain. 



The subjects of this Report may now be stated to be, the 

 leading hydrostatical and hydrodynamical problems recently 

 discussed, which proceed upon the supposition of an incom- 

 pressible fluid, or of a fluid in which the quotient of the pres- 

 sure divided by the density is a constant ; and the end it has 

 in view is, to ascertain to what extent, and with what success, 

 analysis has been employed as an instrument of inquiry in these 

 problems. I am desirous it should be understood that I have 

 not attempted to make a complete enumeration either of the 

 questions that have been discussed in this department of science, 

 or of the labours of mathematicians in those which have come 

 under notice. It has rather been my endeavour to give some 

 idea of the most approved methods of treating the leading 

 problems, and the possible sources of error or defect in the 

 solutions. In taking this course I hope I may be considered to 

 .have acted sufiiciently in accordance with the recommendation 



