VOL. LI.] PHILOSOPHICAL TKAKSACTIONS. oQQ 



base IK, of the rectangle triangle ikn, may be equal to ab ; and the perpendi- 

 cular NK = BE ; let the plane in be struck by the wind, in the direction lm, 

 perpendicular to ik: then, according to the known rules of oblique forces, the 

 impulse of the wind on the plane in, tending to move it according to the direc- 

 tion LM, or NK, will be denoted by the base ik; and that part of the impulse, 

 tending to move it according to the direction ik, will be expressed by the per- 

 pendicular NK. Let the plane in be moveable in the direction of ik only; that 

 is, the point i in the direction of ik, and the point n in the parallel direction nq. 

 Now it is evident, that if the point i moves through the line ik, while a particle 

 of air, setting forwards at the same time from the point n, moves through the 

 line NK, they will both arrive at the point k at the same time; and consequently, 

 in this case also, there can be no pressure or impulse of the particle of the air 

 on the plane in. Now let lo be to ik as bp to be; and let the plane in move 

 at such a rate, that the point i may arrive at o, and acquire the position oq, in 

 the same time that a particle of wind would move through the space nk : as oa 

 is parallel to in, by the properties of similar triangles, it will cut nk in the 

 point p, in such a manner, that np = bf, and pk = fe: hence it appears, 

 that the plane in, by acquiring the position oa, withdraws itself from the action 

 of the wind, by the same space np, that the plane ab does by acquiring the po- 

 sition fg; and consequently, from the equality of pk to fe, the relative impulse 

 of the wind pk, on the plane oq, will be equal to the relative impulse of the 

 wind FE, on the plane fg: and since the impulse of the wind on ab, with the 

 relative velocity fe, in the direction be, is represented by-fAB; the relative im- 

 pulse of the wind on the plane in, in the direction nk, will in like manner be 

 represented by ^ik; and the impulse of the wind on the plane in, with the rela- 

 tive velocity pk, in the direction ik, will be represented by -I-nk; consequently 

 the mechanical power of the plane in, in the direction ik, will be -4. the paral- 

 lelogram la: that is -^ik X ^nk: that is, from the equality of ik = ab and 

 NK = BE, we shall have -fia = 4-ab X -fBE = -|-ab X -I-be = -|- of the area of 

 the parallelogram af. Hence we deduce this. 



General Proposition. — " That all planes, however situated, that intercept 

 the same section of the wind, and having the same relative velocity, in regard 

 to the wind, when reduced into the same direction, have equal powers to pro- 

 duce mechanical effects." — For what is lost by the obliquity of the impulse is 

 gained by the velocity of the motion. Hence it appears, that an oblique sail is 

 under no disadvantage in respect of power, compared with a direct one ; except 

 what arises from a diminution of its breadth, in respect to the section of the 

 wind : the breadth in being by obliquity reduced to ik. 



The disadvantage of horizontal windmills therefore does not consist in this ; 

 that each sail ; when directly exposed to the wind, is capable of a less power, 



VOL. XI. 3 B 



