CAPILtART TtJBES. 257 



and of which the base should be the part of the plane com- Laplace's theo* 

 prized between the two horizontal lines drawn through those [|^° j^'j^^^^^^^^^* 

 points. pression of 



In general, if we compare the theory which I expose, to the eff c^t of attnt«- 

 immerous experiments of philosophers on the capillary action, tion which h 

 we shall see that the results obtained by those experiments are f^^f capii- 

 deducible not by vague and uncertain considerations, but by 

 a train of geometrical reasoning, which appears to me to leave 

 no doubt of the truth of this theory. I am desirous that this 

 application of analysis to one of the most curious objects of 

 natural philosophy may prove interesting to geometers, and 

 excite them to multiply more and more these applications, 

 M'hich unite the advantage of giving certainty to physical 

 theories, and adding to the perfection of the analytical art, by 

 the frequent demand for new artifices of calculation* 



Note (by the Author). 



The demonstralion of the preceding theories will be published 

 in one of the succeeding volumes of the Institute. The following- 

 results of analysis may serve to direct those who may be disposed 

 to deduce the principal themselves. 



Let us denote by <p (./") the law of the attraction of a fluid par- 

 ticle upon another particle, placed at the distance/; ^ {/) de^ 

 creasing with an extreme rapidity, while /"increases, and being 

 insensible for every sensible value of /. Let us also designate by 

 c — n (/) the integral / df. (p (f) taken from/=o, c being the 

 value of that integral, when/ is infinite ; n (/) will in like maa-* 

 ner decrease with an extreme rapidity, and will be also insensible 

 for all the sensible values of/. Let us also denote by c' — f if) the 

 integral// d/. IT (/), c' being its value when /is infinite; f (/) 

 will be likewise insensible for ail the sensible values of/. Lastly, 

 let us denote by K and Hthe integrals 2 'rrfd %. "f (z) and 2 t/z d 

 X. f (z) taken from z nul to z infinite, v being the semi-circumference 

 of which the radius is unity. It will be seen by the analysis of No. 

 12, of the second book of La Mecanlque Celeste, that the action of 

 a sphere of which the radius is b, upon the fluid included in a canal 

 infinitely narrow, perpendicular to its surface, is K-{- r- By this 

 action I understand the pressure which the fluid of the canal would 

 exert by virtue of this action, upon a base perpendicular to the 

 direction of the canal, placed in iti interior at any sensible distance 

 whatever, from the surface of the body, and taken for unity. This 

 would also be the expression of the action of a body terminated by 

 a sensible segment of a sphere whose radius is 6; which results 



Vol. XIV.— July, 1806. LI ftvi». 



