OF FLUID PRESSURE ON CIRCULAR PLANES AND ON SPHERES. 95 



106. The preceding investigation applies to the particular case, in 

 which the pressures on the curve surfaces of the segments are equal 

 to one another ; but in order to render the solution general, we must 

 investigate a formula to indicate the point of division, when the 

 pressures are to one another in any ratio whatever ; for instance, that 

 of m to n. 



By expunging the common factors from the equations (51) and 

 (52), we obtain 



a : 2 { (D a) x + } (D *) 2 } : : m : n ; 



therefore, by equating the products of the extreme and mean terms, 

 we get 



2m{(Dx)x + J( D xY} = nx*, 

 which, by expanding the bracketted expression, becomes 



nxt m^ x*}, 

 or by transposition, we obtain 



(m -\- n) x* nz m D 2 , 

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



rm 

 m+n (54). 



107. The general equation just investigated, is sufficiently simple 

 in its form for every practical purpose that is likely to occur ; it may 

 therefore appear superfluous to reduce it to a rule, yet nevertheless, 

 that nothing may be wanting for the general accommodation of our 

 readers, we think proper to draw up the following enunciation. 



RULE. Divide the first term of the ratio by the sum of the 

 termSj and multiply the square root of the quotient by the 

 diameter of the sphere ; then, the product thus arising, will 

 express the distance below the surface of the fluid, of that 

 point through which the plane of division passes. 



It is unnecessary to propose an example for the purpose of illus- 

 trating the above rule ; that which we have already given, where the 

 values of m and n are equal to one another, being quite sufficient. 



