104 OF FLUID PRESSURE UPON THE ANNULI OF A CYLINDER. 



sustitute in this equation, the values of x, x and x" as represented in 

 the equations marked (63), (64), and (65), and we get 



(2 -t/2) V D ^-KV6 2) V 

 and this, by a little further reduction, becomes 



therefore, by completing the square, we obtain 



a;"'* 4- i/fiDd.x" 1 -f i Drf 2od; 



and finally, by extracting the square root and transposing, we have 

 x 1 " = J (2 V~2 V 6) V^ (66). 



112. And thus we may proceed to any extent at pleasure ; that is, 

 to any number of annuli within the limit of possibility ; for it is mani- 

 fest, from the nature of the problem, that impossible cases may be 

 proposed, but the limit can easily be ascertained in the following 

 manner. 



It is obvious, that the sum of the breadths of the several annuli, is 

 equal to the whole depth of the vessel ; and that the sum of the 

 pressures is equal to the pressure on the concave surface ; but in the 

 problem immediately preceding, we have demonstrated that the pres- 

 sure on the bottom of a cylindrical vessel, is to that upon its upright 

 surface, as the radius of the base is to the perpendicular altitude. 



Now, according to the conditions of the question, the pressure on 

 each annulus is equal to that upon the base ; consequently, in order 

 that the problem may be possible, the depth of the vessel must be 

 equal to the radius of the base, drawn into the number of annuli. 



If instead of D the diameter of the cylindrical vessel, we substitute 

 2R its equivalent in terms of the radius, the preceding equations (63), 

 (64), (65), and (66), will become transformed into 



x = (/f v'O) y^, (67). 



*' (V2 /f) Va3, (68). 



~ " (69). 



(70). 



From these equations the law of induction becomes manifest, and 

 the general expression for the breadth of the n^ annulus, is 



(71). 



