56 HORIZONTAL SECTIONS SUSTAINING EQUAL FLUID PRESSURES. 



Again, by comparing equation (30), with the second of the fore- 

 going expressions for the value of p, we shall have 



bl* 

 bw(v + \w}= - 



and substituting the value of v in terms of / and n, we obtain 



n w* -f- 2/-V//T7 w P, 

 from which, by dividing by n, we get 



w -- . w =. > 

 n n 



therefore, if this be reduced by the rule which applies to the resolution 



of adfected quadratic equations, we shall obtain 



I _ ___ 



w :rr -( ^/2w V n )' 

 n 



Proceed as above, by comparing the equation (30), with the third 

 of the preceding expressions for the value of p, and we shall have 



= 



and if the above values of v and w, as expressed in terms of I and w, 

 be substituted instead of them in this equation, it will become 



and dividing by n, we get 



n n 



therefore, by completing the square, evolving and transposing, we 

 obtain 



By pursuing a similar mode of comparison, and reasoning in the 

 same manner, with respect to the fourth value of p foregoing, we 

 shall have 



let the values of v, w and x, as determined above, be respectively 

 substituted in this equation, and it becomes 



complete the square, and we obtain 



