448 PHILOSOPHICAL TRANSACTIONS. [aNNO 1767' 



§ III. If the wheel be plunged as deep as its axle, that is, /= 0, the equa- 

 tion is changed into this ^ nbaa (-^tw — ^vz -\- -J-«z) = d-Tr, where it appears l". 

 That the quantities d, tt, v and z remaining the same, we have b inversely pro- 

 portional to the square of a ; whence it follows, that if the length i is to be di- 

 minished, without altering the effect of the float-board, the height a must be 

 increased proportionally to the square root of b ; for example, if b is to be made 

 4 times less, it will be sufficient to double the height a. 1° That the velocity of 

 the float-board remaining still the same, the weight tt will be in the compound 

 ratio of the length b, and of the square of the height aa. 3" The dimensions of 

 the float-board remaining, the more the quantity z is increased, the more must 

 the weight tt be diminished. If z be made = O, we have 



TT = TT-,. -i-ff, and if z ■= t; we havcir = — -j. -t^vv, that is 6 times greater than 

 in the first case; which is very conformable to the nature of things; for when 

 the wheel is in motion, the stream then not acting on it but with the excess of 

 its velocity above that of the wheel, it follows, that the greater such velocity is, 

 the more will the effect of the stream be diminished. 



It follows from our last remark, that the greatest weight with which the stream 



can constitute an equilibrium, will be = " . ; but then the wheel will not 

 have any motion, nor consequently the weight tt: if the float-board be increased, 

 or the weight diminished, from that instant the wheel will begin to turn, and the 

 swifter as the float-board is greater, or the weight less; but in most machines, 

 it is- required that the weight may be the greatest possible, as also the velocity 

 with which it is raised. A question therefore here offers itself, whose solution 

 is of much importance. What must be the velocity of the float-board whose di- 

 mensions are given, that the product of the weight by its velocity shall be the 

 greatest possible? 



§ IV. The velocity of the weight tt is - z feet in a second, which being mul- 

 tiplied by the value of tt = '^ {\w — ^ vz + -J-zz) gives the product ^v nba (-i- 

 v^z — ^uzz •\- ■r'^), which must be a maximum; for which purpose make -^wdz 

 — ^vzdz -\- \z^dz = O; whence we have z = ~ = 0.53752f : this value 

 of ^ being substituted, makes the equation -ivi; — \vz -\- -i-zz = — iv, 



so that we have the equation baa = rr^ f- -^ = 280.529 -|- jj^. 



which expresses the dimensions of the float-board when the effect will be the 

 greatest possible. If the float-board be plunged no deeper than to cc, as we 

 have at first supposed, the most advantageous value of z may be determined in 

 the same manner, which will be found to be 



