80 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



but if the operation be performed according to the rule derived from 

 the analogy of comparison, we shall have 



p = f P ; that is, p \ x 16* X 40 X 62 1 X = 266666f Ibs. 



6. THE METHOD OF DETERMINING THE POSITION OF THE CENTRE 

 OF GRAVITY OF THE SPACE COMPREHENDED BETWEEN THE 

 PARABOLIC CURVE AND ITS CIRCUMSCRIBING PARALLELOGRAM. 



94. By means of the pressure on the semi-parabola, as we have 

 investigated it in the two foregoing cases, we are enabled to determine 

 the pressure on the space CFB, and from thence, the position of its 

 centre of gravity. 



This is an important inquiry in the practice of bridge building, for, 

 in determining the thickness of piers necessary to resist the drift or 

 shoot of a given arch, independently of the aid afforded by the other 

 arches, it becomes requisite to find the centre of gravity of the span- 

 drel or space BFC, which is used for the purpose of balancing the arch 

 and filling up the haunches or flanks. 



Now, the method which has generally been employed for the deter- 

 mination of this centre is extremely operose, and in many cases it 

 involves considerable difficulty, requiring the calculations of solids 

 and planes, which are by no means easy ; but the method which we 

 are about to employ, requires no such tedious and prolix operations, 

 as will become manifest from the following investigation, which refers 

 to the space comprehended between a semi-parabola and its circum- 

 scribing rectangle. 



Let BCD be a semi-parabola, having the axis DC vertical while the 

 corresponding ordinate DB is horizontal, and 

 let B DC F be the circumscribing rectangular 

 parallelogram. 



Suppose the point D to remain fixed, and 

 conceive the semi-parabola BCD to revolve 

 about the point D until it comes into the 

 position A ED, where the axis DE is horizon- 

 tal, and the corresponding ordinate DA vertical; then it is manifest, 

 that the circumscribing rectangular parallelogram ADEH in this 

 latter position, is equal to BDCF in the former, and consequently, 

 the space AHE comprehended between the sides of the rectangle 

 AH, HE and the curve AE, is equal to the space BFC similarly con- 

 stituted. 



