132 



THE CIVIL EKGINEER AND ARCHITECT'S JOURNAL. 



LApiilL, 



<juantity more frequently snui;ht. 

 charged in a second: we have 



Q = n'V> 



or v= 1-273 



Let Q be the volume dis- 



Q 



D- 



(VI.) 



This value of r, put in the equation above, transforms it to 



[Inmetr.] II— -0820* y^^ =-002221 —(Q-+ -0432^0-)) 



, >... (VII.) 



[In feet] II--025I9^-i =-000677j^^(Q-+-U173QD-)^ 



Such is the formula which we shall liave to employ for the solu- 

 tiiMi of questions relative to the motion of water in conduit pipes; 

 atteiulirip always, in its practical application, to tiie observations 

 iihich will hereafter follow. Of the four quantities Q, U, H, and 

 L, three being given, the fourth may be found by this formula. 



7. When the velocity is great, so as to exceed 2 feet per second, 

 the resistance is sensibly proportional to the square of the velocity; 

 the term in which it is but the first power disappears, and we have, 

 according to the experiments of Couplet, 



[In metres] H — -051 ?;= = -001435 "' ^ 



[In feet] H —-0155 i^ = -0004373 



Or, in terms of Q, 

 [In metres] II — -08264 §, = -002320 " 



1) 



hv" 

 D 



...(VIII.) 



[In feet] 



H - 



D' D' 



02519 >-, = -000709 — ^ 

 D' D^ 



D' f 



(IX.) 



It will be borne in mind that the second member of the above 

 equations is the value of the resistance arising from the sides of 

 the conduit. 



b. Disengaging the value of Q from the general equation, it 

 becomes 



^\n 



KX.) 



-0216 LD- / 4MI- 2 HD^ / -0216 LD-\- 



[I.. metr.] Q= — L:r3r-2l3+ V L + 3/2U + \LT37TuJ 



r, c .-, rv -0709LD- /li7;0GlID- / -2325 LD-\-' [ 



['"f-<] Q--L.3;-2D^V L.372D^(L^3yTn) J 



In long conduits, where 37 D is very little compared with L, we 

 may neglect it; and again neglecting the second term under the 

 root, we shall have for ordinary cases of practice. 



VHD- 

 — J— 



[In feet] Q = 38- 1-305 t/ ■ 



'^HD^ 



•021 G D= 



— -0709 D- 



f 



...(XI.) 



or, Q = 20-3 



.(XII.) 



9. In great velocities, it is 



I1IJ = 



[In feet] Q = 37-034a/ ''"!. >' 



If the velocity is required, we obtain its value by dividing the 

 (juantity Q by the section tt'D-. 



Expression fur tlie Diameter. 



10. The diameter of conduits is very often the quantity we have 

 to determine. The best method of obtaining it is by putting the 



„ „,.. /hd 



or, Q = 3C-77a/ — 



fundamental equation under the following form 



tin metrts] D 3 -{ -00009594 -g- D2 + -0826 -g- D + -tm'J2 ~ 



, I.Q Q" LQ2 , I 



fill feet] DS- {•00003594 -Jj- D2 + -02:iHjj- D + -OOOG77-^- I =0 



We may pass over, for a first approximation, the first two terms 

 in the brackets, and we have. 



=o1 



!>.(xiii. 



fin metres] D = a / 

 [In feet] D =/// •> 



00222 



LQ- 

 H 



•295 



0006769 



LQ- 

 H 



•2323 



'^iL HXIV.) 



IS /LQ^ 



V H 



This value will be rather small; and we must successively make 

 slight augmentations to it, until the first member of the equation 

 is reduced to, or equals. 0. The quantity which shall have led to 

 this result will be the diameter required. 



For velocities above 2 feet per second, we may take directly and 

 simply 



[In metres] 

 [In feet] 





..(XV.) 



I need say nothing on the determination of H and L; the equa- 

 tion (VII.) gives them by a simple transposition. 



11. Let us take some examples of the determiDation of discbarges and 

 diameters : — 



Ex. 1.— We have a conduit of (25 metr.) •820225 feet diameter, and 

 (1450 Dietr.) 4757-3 feet l{ing: required the volume of water it will dis- 

 cbarge per second, with a head of (5-32 metr.) 17-454 feet ? 



We have here D = -820225 feet ; H=. 17-454 feet ; I. = 4757-3 feet ; and 

 L + 37-2D = 4787-8l6 feet. Consequently (X.), 



•0/09 X (•8'20225;2 X 4757-3 

 Q=- 



47«7-H18 



v- 



i; -154 X (■S20225;i /••2.i2S x 4r67-,1 x -WM-ilWi 



47S7-S16 



^ 



47»7-»l« 



= -•04737G + ^/iy98y + -024155 



= -047376 + 1-423 = 1-37924 cubic feet per second, 



the quantity required (all the measures being in English feet). The simpli- 

 tied formula (XI.) would have given 



Q=l-4185--047G7 = l-3708 cubic feet. 



That for great velocities (XII.), and applicable to this ca-te, in which the 

 velocity is 2-6 feet per second, would have given 1-357 cubic feet. 



Ex. 2. — Required the diameter of a conduit, 2483-64 feet (757 metr.) 

 long, and whiih shall convey 3-14317 cubic feet (-089 metr. cub.) per 

 second, with a head of 3-2809 feet (1 metr.)? 



Putting these numerical quantities in the equation (XIII.), it becomes, all 

 reductions made, 



D= -(-22827 D^ + -07581 1 D + 5-0G04) = 0. 



Neglecting the first and second terms, we have 



D = v'5-0604 = 1-383 feet. 



Tliis value of D, put in equation (XIII.), will be found too small; by 

 gradually increasing it, we shall find, by a few trials, that the value 1-4127 

 feet for D, will reduce the first member of the equation to 0, and will be the 

 diameter sought. 



The formula for great velocities (XV.), and in this case v exceeds 2 feet 

 per second, would have given 



D=-233^ 



24 83 64 X (3-143) 2 

 3-2809 



1-383 feet. 



[\Ve shall next month proceed to the author's consideration of 

 conduits terminated by adjutages.] 



REVOLVING ELLIPTICAL WHEELS. 



Sir — Having had occasion to seek for some simple means of 

 producing a variable motion of rotation round one fi.xed axis from 

 a unifoi-m motion round another, I have been led to observe a 

 property of the ellipse, which as it was new to me, may perhaps 

 prove so likewise to some, at least, of your readers. 



It is, that if two equal and similar cogged wheels of elliptical 

 form, be geared together as represented in the annexed figure 

 (which is a drawing of the pitch lines of such wheels, without the 

 cogs), the teeth will continue to act upon one another during an 

 entire revolution, with perfect regularity; and the motion of the 

 one axis will be transferred to the other — not uniformly, hut sub- 

 ject to a variation in velocity, the nature and amount of which 

 may be easily calculated. By such an arrangement, therefore, a 

 variable motion may be produced from a uniform one, in a manner 

 comparatively simple and easilj' available, — capable of transmit- 

 ting a force of any amount with certainty and precision. There 

 are, probably, many cases in which some such arrangement would 

 be found convenient; and I am inclined to believe that it is not 

 possible to find any more simple means of attaining the object. 



The conditions which must be fulfilled in order that any two 

 curves — supposed to act in the manner represented, from fixed 



r 



