ON THE SAILS OF WINDMILLS. '395 



to WV; then WC will be the relative velocity and proporti- 

 onate force with which the wind strikes the vane perpendicu- 

 lar to its plane, which force being resolved into two forces WC 

 and cC, the first perpendicular to the plane of motion and 

 therefore of no effect, the last parallel to it and therefore re- 

 presents the effective force of every particle of wind to turn 

 the vane, when the force perpendicular to the plane is CW ; 

 now let any other angles of weather as VSB, VSA, be taken, then 

 will WSB, WSA be the relative angles of incidence ; BW, AW 

 the relative velocity and proportionate force perpendicular to 

 the plane of the sail ; and Bb, Aa the effective forces to turn 

 the vanes SB, SA ; whence it appears that these forces are or- 

 dinates to the chord WV, and if the force of the wind on the 

 vane were in the simple ratio of its relative velocity, or the num- 

 ber of impinging particles were invariable, as is the case in un- 

 dershot water wheels, as observed by Mr. Waring, in the third 

 volume of the Philosophical Transactions ; then the greatest 

 force would be SB, where the angle of weather VSB equally di- 

 vides the angle of limit WSV, which agrees with the theorem 

 given by Maclaurin and Simpson, on the above supposition, 

 which to distinguish we may call " Warings hypothesis." But 

 if the force be as the square of the relative velocity ; describe a 

 semicircle on SW, and with the radius SD describe the arc D 

 RE cutting the plane SC in R from which draw RF, RG per- 

 pendicular to SW, SV. Then the proportionate force on any 



WC 2 

 point C perpendicular to the plane SS will be ; or (since R 



RF 2 WV 2 



F, SN are halves of CW, WV,) ; and as WC : cC or SR : 



SN 2 RF 2 RF 2 XRG 



RG : : (the force perpendicular to SC,) : 



SN 2 SN 2 XSR 



the whole effective force on each point of the plane SC, which 

 is the same as given by Maclaurin and Simpson, and agrees 

 with their hypothesis. 



But the same breadth of sail inclined to different angles of 

 weather will not intercept an equal current of wind ; the rela- 

 tive current being the parallelogram WZCP to the plane CS, 



