4(^4 InJivuBhns for meafnringthf Force of a Stream y and determining- 



In the confideration of power or forcf to be derived from water in motion, thi water 

 may be taken as a determinate mafs falling through a given height in a given time. In 

 order that this defcending weight may caufe another weight to afcend, or may overcome 

 fome refiftance in the way of work with that degree of fpeed which fliall be the moft. pro- 

 fitable, it is neceflary that the refiftance or work to be done fliould be neither too great nor- 

 too little. If it be too great, the flownefs ef operation will diminifii the quantity of work ; 

 and if it be too fmall, the fpeed will not fufficiently compenfate for this fmallnefs. When 

 the power is therefore known, it remains to deduce what may be the cffe£t. But in the 

 firft place, as the height from which the water flowing in a river may have defcended, in 

 order to acquire its velocity, is, from a variety of circumftances, difficult to be afcertained, 

 and alfo very different from that height which would Immediately and without impediment 

 produce the fame velocity, it becom.es neceflary to compute this laft height, which hydro- 

 ftatical writers ufually call the height of the virtual head. Thefe writers teach, that the 

 velocity of a fluid fpouting through an orifice in a thin plate is the fame as would be ac- 

 quired by a body falling in clear fpace from the height of the furface of the fluid above the 

 ©rifice. Hsnce, from the common doftriue of falling bodies, if the unlforrri velocity of a 

 ftream be exprefled in feet per fecond, the virtual height of the fall will be found by mul- 

 tiplying the given velocity into itfelf, and dividing the produ£l by 64,2882; the quotient 

 will be the required height eJf'prefled in feet *. 



The cfFeft of underftiot and overfhot-wheels has been treated by various authors, who 

 have given refults extremely different from each other. Smeaton, in the Treatife often^ 

 quoted in the courfe of this communication, obferves, that Belidor in his Archite£lure Hy- 

 draulique, 1. 286, endeavours to demonftrate that water applied uhderfliot will do fix 

 times more execution than the fame applied overlhot ; while Defaguliers, whom he 

 (Smeaton) mifquotes by overlooking the difference of fail. Is faid to have given the advan- 

 tage as ten to one in favour of the overlhotf. The particular experiments of Smeaton- 

 himfelf, as well as his experience, point out the following refults. 



The effefl; in underfliot-mills in the large way is at beft one third' of the power ; that is- 

 to fay, the wheel, being driven with two-fifths of the velocity of the ftream, will raife a. 

 quantity of water equal to one-third cf the column, which ftrikes the float-boards, to an 

 height equal to that of the virtual head or fall : and the effedl of an overfliot- wheel will 

 be, at a medium, twice that of the underfliot. Mills having a breaft-wheel, or other kind 

 of wheel on which the water ads, partly by its weight and partly its impulfe, will produce 

 more or Icfs effed, accordingly as the circumftances approach more nearly to thofe of the 

 over or underlhot-wheels. 



For the advantage of fuch as are leaft converfant in fubjeifls of this nature, for whom 

 chiefly the prefent memoir is intended, I ftiall illuftrate the fubjedt by an example. 



Suppofe a ftream to pafs through an eftate without any evident fall, with' a velocity of 

 nine feet per fecond, and affording fuflScient room to place an underftiot-wheel with a pro- 



* The f rcof of this is fimpU-.but may alfo be feen in Delagiiliers, ii. 510.— Thofe who ufe logarithms may 

 with Icfb trouble /ai/rafl tie conjiant log. i.icSi^\2 from Itvlce tit log. nf tkt veheily, and the remainder wili be the Itg. 

 if the virtual height. 



f The proportion of work of the two aftual mills, compared by Defaguliers, is as 3,25 to 1 in favour of the 

 «\?iiliot. 



