TOL. tVII.} PHILOSOPHICAL TRANSACTIONS. 453 



^ X. If we take nothing but the most advantageous position into considera- 

 tion, and preserve the greatest effect entire, it follows that the angle ecd, fig, 

 13, between two float- boards, must be such, that e should enter the water at 

 the instant when ab quits the vertical, so that the cosine of that angle be 

 = -; in consequence of which the following table may be constructed, showing 



what the number of float- boards should be, for a given ratio between /and a. 

 For 4 float-boards, we have/ = 0. For 10 float-boards, we have/ = O.SOpOa. 



5 0.3090a. 12 0.8669a. 



6 0.5000a. 14 0.9009a. 



7 0.6236a. 16 0.9239a. 



8 0.707 la. 18 0.9397a. 



9 0.7660a. 20 0.9510a. 



&c &c. &c &c. 



^ XI. Certain authors, treating of hydraulics, have in this part of it given 

 the same table, as containing the true number of float-boards the wheel should 

 consist of; but we have seen on what principle it was formed, and that it was 

 only to preserve entirely the effect of the vertical float-board; whence it follows 

 not that the number of float-boards which it assigns should be the most advan- 

 tageous. To which purpose the effect produced from every position of the wheel, 

 and for the different number of the float-boards, should be computed ; the number 

 which gives the arithmetical mean between all these effects, the greatest of all, 

 will be that to be chosen, and preferred before what the above table indicates. 

 It may be sufficiently satisfactory to compute only the effect from 1 to 10 degrees: 

 thus, for example, for the wheel entirely plunged we are to find the effect, fig. 

 J 4, 1° on OA, 2° on oi and gb, 3° on oh, and fc, 4° on OG and hd, 5" on op, 

 and pe, 6° on oe, 7° on od, 8° on oc, and 9° on ob. After which the wheel 

 returns into the same position it had at first; and we are to divide the sum of all 

 these effects by Q, to get the arithmetical mean. 



We will next suppose the number of 6 float-boards for the same case of y= O, 

 and compute the following effects : l°on og -\- ax, 3° on oe -|- nc, 5° on oi -f- 

 yc, 2° on OF -|- mh, 4° on od -|- nc, 6° on oh -|- ^b. The sum of all these 

 effects divided by 6 will give the effect of the wheel of 6 float-boards. 



The same thing, supposing the angle 40 degrees, or Q float-boards, and as 

 after a revolution of these 40 degrees, the wheel returns into a similar position, 

 the same must be divided by 4. Then for an angle of 30 degrees we are to di- 

 vide by 3, and so on. 



Mr. M. made this computation to great exactness, for the case ofy=0, 

 /= \-a, and/ = 0.866 a = a cos. 30°; the result, 1° '\{ J — O, for 4 float- 

 boards, the arithmetical mean = 0.335 {^nbaavv). 



It may be observed in this first case, that there is some advantage in taking 6 

 float-boards instead of 4, shown by the table; the effect will be increased in the 



