82 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



LMabch, 



be 11 feot, and by tlie table t!ie parabola A O R = 1516 68 ; the 

 other, A II V, 4 feet bij.^li, will be 332'6t; whence the trapezium 

 H Y R () will be the scale of the velocity, ami tlie ((uaiitity of 

 water passim; in a yiven time t'irou:;h the section II () ^ IlKt'Ot. 

 if the parabola S I' () be .'5 feet liigh, its value in the table = 

 216'0U; then the parabolic trapezium t^ K Y H, beina: eciualtothe 

 aforesaid parabola S 1* t), will be SIO'OO, which substituted from the 

 total value of A H Y, there remains the parabola A Q K = 116-61.. 



I his numl)er not beina: precisely to be found in our table, find the 

 ne.\t highest, =: U7-60, wliich corresponds to a height of 2 feet; 

 whence we arrive at tiie conclusion tliat the regurgitation at the 

 point O has raised the water 2 feet more than the first, supposed 

 to be 4 feet." 



To facilitate the practical application of the principles con- 

 tained in Grandi's proposition, the following rules will be found 

 convenient : — 



The height and width of the section of both the influent and 

 the recipient being given in each case and their velocity being equal. 



1. When the velocity of the united streams is the same with 

 that of each separately, to find the increased height of the united 

 section. 



Find in the table the parabolic value in the third column corre- 

 sponding to the given height of the recipient in the first. Multiply 

 this value by the given width. Perform the same operation for 

 the influent, we shall then have obtained the quantity of water 

 brought down by each. Add these two quantities together. Divide 

 their sum by the width' of their united section, which may be 

 either that of the influent, or of the recipient, ot greater or less 

 than either. Find the quotient obtained by such division in the 

 tlurd column of tlie table, opposite to it in the first will be found 

 the height of the united sections. 



2. When the velocity of tlie united streams is increased, to find 

 the height of their united section. 



Divide the height found by the preceding rule by the number of 

 times by which the velocity is increased, the quotient is the height 

 of the united sections. 



3. When tiie velocity of the united streams is diminished, to 

 find the height of their united section. 



Multiply the height found by our first rule by the number of 

 times by which the velocity is diminished, the product gives the 

 required height. 



4. When the height of the united streams remains the same, to 

 find their increased velocity. 



Divide the height as found by the first rule by the original 

 height, the quotient will give the increased velocity. 



5. When the height of the united streams is increased, to find 

 their velocity. 



Divide the height found by the first rule by the increased height, 

 the quotient gives the diminished velocity. 



6. AVhen the height of the united streams is diminished, to find 

 their increased velocity. 



Divide the height found by the first rule by the diminished 

 height, the ipiotient will be the increased velocity. 



To exemi)lify these rules a small table is subjoined, constructed 

 from Grandi's data, that is, supposing a stream 760 feet wide and 30 

 feet high to receive successively 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 

 similar influents. The first column contains the number of influ- 

 ents ; the second, the height caused by the addition of these suc- 

 cessive streams as calculated by our first rule, that is, supposing 

 the velocity to remain the same ; the third column shows the in- 

 creased height found by Gennete, the original height, 20 feet, being 

 here increased by the addition of ,j, ,1, fj, &c. The fifth column 

 shows the increased velocity requisite to produce the height shown 

 in the third ; thus supposing a stream 760 feet wide and 30 feet 

 high to receive two other similar streams, the increased height, 



according to Gennete, will be 30 feet 7'6 inches, and to produce 

 such a height the required velocity will be r97233. Either of 

 these numbers is deducible from the other by one of the preceding 

 rules; thus, sujiposing the height 30 feet 7 '6 inches to be given, 

 and the velocity to be required, by Rule 5, dividing 62 feet i'G 

 inches by 30 feet 7-6 inches we obtain a quotient of 1-97233. 

 Suiiposing, on the other hand, the velocity 1-97233 to be given, we 

 olitain the height by Rule 2, since 62 feet 4-6 inches -;- 1-87233 

 = 30 feet 7-6 inches. The fourth column shows the increased 

 velocity required to maintain a constant height of 30 feet, and is 

 found bv Rule 4. 



It is found that the several increments of either height or velo- 

 city are as the ordinates of a parabola whose axis is divided into 

 the same number of parts as there are required velocities. Hence 

 an elegant method of finding the intermediate heights or velocities 

 when the two extremes are given. Suppose, for example, w-e 

 require to find the several heights indicated in our first column. 

 Find tlie height required for twelve streams by our Rule 1, Draw 

 A B, and from a scale of equal parts set oS' 157 feet 3 inches from 



A to C, at A erect a perpendicular A D to A B, and set off twelve 

 equal parts thereon, and draw through the points I, 2, 3, &c., 

 lines parallel to A B, on the parallel I E, set otf the first height 

 30 feet from the same scale as A C. Then by Rule 1 find the 

 height of any one of the intermediate streams, as 6, and set it oflF 

 from 6 to F, then through the points E, F, C, describe a parabola, 

 the portion cut ofi' on each ordinate by the curve will be the 

 several numbers given in the table as measured by the scale from 

 which I E, 6 F, and A C were taken ; the abscissas 1, 2, 3, &c., 

 may be set ofi" by any scale, providing they are equidistant, and 

 according as they are wider or narrower, will the parabola in- 

 crease or diminish its curvature. It is evident that in the case of 

 100 additional streams the labour of calculation will be materially 

 shortened, as no more than three values need ever be found arith- 

 metically. 



In like manner either of the other values shown in our table 

 may be represented parabolicallv. Column 5, for example, by 

 setting otf 1-34203 on I E, 4-33793 on A B, 3-04897 on 6 F, and 

 describing a parabola through those points. 



