PHILOSOPHICAL TRANSACTIONS. 



359 



VOL. LI.] 



If nothing more was intended than to determine the most efficacious angle to 

 make a mill acquire motion from a state of rest, or to prevent it from passing 

 into rest from a state of motion, we shall find the position of N° 1 the best ; 

 for if we consult col. 7? which contains the least weights that would make the 

 sails pass from motion to rest, we shall find that of N° 1, (relative to the 

 quantity of cloth) the greatest of all. But if the sails are intended, with given 

 dimensions, to produce the greatest effect possible in a given time, we must en- 

 tirely reject those of N° 1 ; and if we are confined to the use of planes, con- 

 form ourselves to some angle between N° 3 and 4, that is, not less than 72*^, 

 nor greater than 75°, with the axis. 



The late celebrated Mr. Maclaurin has judiciously distinguished between the 

 action of the wind on a sail at rest, and a sail in motion; and, in consequence, 

 as the motion is more rapid near the extremities than towards the centre, that 

 the angle of the different parts of the sail, as they recede from the centre, should 

 be varied. For this purpose he has furnished us with the following theorem.* 

 * Suppose the velocity of the wind to be represented by a, and the velocity of any 

 given part of the sail to be denoted by c; then the effort of the wind on 

 that part of the sail will be greatest, when the tangent of the angle in which the 



wind strikes it, is to radius, as \/ 2 -^ -^ -\- — to I J' This theorem then assigns 



the law, by which the angle is to be varied according to the velocity of each part 

 of the sail to the wind : but as it is left undetermined what velocity any one given 

 part of the sail ought to have in respect to the wind, the angle that any one part 

 of the sail ought to have, is left undetermined also; so that we are still at a loss 

 for the proper data to apply the theorem. However, Mr. S. being willing to 

 avail himself of it, and considering that any angle from 15'^ to 18° was best suited 

 to a plane, and of consequence the best mean angle, he made the sail, at the 

 middle distance between the centre and the extremity, to stand at an angle of 

 15*^ 41' with the plane of the motion ; in which case the velocity of that part of 

 the sail, when loaded to a maximum, would be equal to that of the wind, or 

 c = a. This being determined, the rest were inclined according to the theorem, 

 as follows : 



• Maclaurin's Account of Sir Isaac Newton's Philosophical Discoveries, p. 176, art. 29. Orig. 



