VOL. LXVI.] PHILOSOPHICAL TRANSACTIONS. 75 



subject. These experiments, with the conclusions drawn from them, have 

 already been communicated to this society, who printed them in vol. 51 of their 

 Transactions for the year 1759, and for this communication I had the honour 

 of receiving the annual medal of Sir Godfrey Copley, from the hands of our 

 very worthy president the late earl of Macclesfield. Those experiments and 

 conclusions stand uncontroverted, so far as I know, to this day ; and having 

 since that time been concerned in directing the construction of a great number 

 of mills, which were all executed on the principles deduced from them, I have 

 by that means had many opportunities of comparing the effects actually pro- 

 duced with the effects which might be expected from the calculation : and the 

 agreement T have always found between these two, appears to me fully to establish 

 the truth of the principles on which they were constructed, when applied to 

 great works, as well as on a smaller scale in models. 



Respecting the explanatory deduction of Desagiiliers, in the first example 

 abovementioned, which indeed I have found to be the commonly received doc- 

 trine among theoretical mechanics, it is shown, Phil. Trans., vol. 51, p. 120, 

 121, and 123, part 1, maxim 4, that, where the velocity of water is double, 

 the adjutage or aperture of the sluice remaining the same, the effect is 8 times ; 

 that is, not as the square but as the cube of the velocity ; and the same is inves- 

 tigated concerning the power of the wind arising from difference of velocity, 

 p. 156, being part 3, maxim 4. 



The conclusion in the 2d example abovementioned, adopted both by Desagu- 

 liers and Maclaurin, is not less wide of the truth than the foregoing ; for if that 

 conclusion were true, only ^r^ihs of the water expended could be raised back 

 again to the height of the reservoir from which it had descended, exclusively of 

 all kinds of friction, &c. which would make the actual quantity raised back 

 again still less ; that is, less than 4- of the whole ; whereas it appears, from 

 table 1, p. 115 of the said volume, that in some of the experiments there related, 

 even on the small scale on which they were tried, the work done was equivalent 

 to the raising back again about J- of the water expended ; and in large works the 

 effect is still greater, approaching towards half, which seems to be the limit for 

 the undershot mills, as the whole would be the limit for the overshot mills, if it 

 were possible to set aside all friction, resistance from the air, &c. 



The velocity also of the wheel, which, according to M. Parent's determination 

 adopted by Desaguliers and Maclaurin, ought to be no more than J- of that of 

 the water, varies at the maximum, in the abovementioned experiments of table 

 1, between 4- and x ; but in all the cases there related, in which the most work 

 is performed in proportion to the water expended, and which approach the 

 nearest to the circumstances of great works, when properly executed, the maxi- 

 mum lies much nearer to ^ than 4-; one half seeming to be the true maximum, 



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