OF FLUID RRESSURE UPON THE ANNULI OF A CYLINDER. 103 



1. 5708 vx*s :=. 

 and casting out the common factors, we get 



2a? ? = vd; 

 therefore, by division and evolution, we have 



x=%^2nd'. (63). 



By proceeding in a similar manner for the breadth of the second 

 annulus, we shall obtain 



and this, by expunging the common terms, becomes 



4*'(* + J*') = Drf; 



therefore, by substituting for x, its value as expressed in equation (63), 

 we shall get 



complete the square, and we have 



a' 2 -\-^2iTdx + %vd=: vd; 

 and finally, extracting the square root and transposing, we obtain 



*' = 4(2^2) J~d. (64). 



Again, by performing a similar process for the breadth of the third 

 annulus, we shall have 



3.1416D3"s (x -f- x' + i") = .7854 Dd, 

 from which, by casting out the common quantities, we get 



4x"(x + x'+%x")=:i>d; 



therefore, by substituting for x and x', their values as expressed in the 

 equations marked (63) and (64), and we shall obtain 



4x" {J Jtod+ J(2 V2) V~^~d+ ix"} = Dd, 

 and this, by a little reduction and proper arrangement, gives 



complete the square, and we obtain 



consequently, by extracting the square root and transposing, we get 



a /; -=JV6 2)^25- ( 65 > 



Pursuing a similar train of reasoning for the breadth of the fourth 

 annulus, we shall obtain 



3.1416D*"'s (x + x' + x" -f iO = .7854Drf, 

 and by suppressing the common factors, we have 



4*"' ( x + x' + x" 4- 1*'") = d ; 



