39& ON THE SAIL 



the particles intercepted will be as CP ; or Rp on the plane S 



RF 2 xRG 

 R, so that Rpx will represent the force on the plane 



SN 2 xSR SR 



SR, but SN : SD (= SR) : : RF : Rp = RFx therefore 



SN 



RF 3 X RG 



the whole effective force on the section SR will be 



SN- 

 which by Simpson's fluxions vol. 2, Prob. 5 page 503, will be 

 a maximum when MN = 4. SN. But if the whole effective 

 force according to Maclaurin and Simpson be asRF 2 xRG, (SR 

 and SN being constant) the maximum will then be when MN 

 4 = SN. 



But supposing the angle of weather known for every pro- 

 portionate velocity of the sail to the wind, it still remains to be 

 determined what that proportion ought to be at the extremity 

 of the sail, as was justly observed by Mr. Smeaton; who, in put- 

 ting Maclaurin's theory to the test of experiment, assumed it as 

 two to one, but he found that by increasing the angle of weather 

 three and six degrees, the effect or product was still increasing, 

 although by increasing the angles of weather at every part equally 

 they became no longer the angles of Maclaurin ; to have made 

 it decisive he should also have taken it as one and an half to 

 one ; and as one to one, the forces from the theory continually 

 increasing, as these ratios diminish. Mr. Smeaton also appears 

 to have made an error by estimating the mean velocity of the 

 wind from the distance of the axis of his rotary machine from 

 the centre of his sails ; the force being as the squares of the 

 velocities, the mean should have been taken at a greater dis- 

 tance ; if this error be corrected, his conclusion that the extre- 

 mity of the sail should move with 2,7 times the velocity of the 

 wind, will probably be altered to less than double, and from 

 theory a slower motion of the sails appears to be highly ad- 

 vantageous. 



The angle of weather on either of these hypotheses is very 

 easily laid down by the following construction, from whence 

 some useful conclusions may be drawn. 



