VV^IND-MILL. 



der, is oblique to the keel. The rudder, then, would be 

 carried otF in an oblique direftioii ; but as, in reality, it is 

 fo fecured, that it cannot be torn and carried off, we are 

 only to confider, in this compound motion, that of the two 

 direftions wherewith it can move without being torn from 

 the keel ; and leave the other, which would tear it off, as 

 ifelefs. 



Now, the direction in which it can move without parting 

 Tom the keel, is that whicli carries it circularly about its 

 ;xtremity, as about a centre. So that the efFeft of the 

 jblique impulfe of the water on the rudder is reduced, firfl 

 a perpendicular impreflion, which is again reduced to the 

 nere turning of the rudder round ; or, if the rudder be im- 

 noveable, to the turning of the veflel. Now, in an oblique 

 ind compound motion, where only one of the direftions is 

 )f fervice ; the greater ratio the other has to it, the lefs 

 iffedl will the motion have, and vice ver/d. In examining 

 he compound motions of the rudder, we find, that the 

 nore oblique it is to the keel, the ratio of the direftion that 

 erves to turn it to the other is the greater. But, on the other 

 land, the more obliquely it is to the keel, and, of confe- 

 [uence, to the courfe of the water which is fuppofed parallel 

 o it, the more weakly it ftrikes. The obliquity of the 

 udder, therefore, has, at the fame time, both an advantage 

 nd a difadvantage ; but as thofe are not equal, and as each 

 if them is ftill varying with every different polition of the 

 udder, they become complicated varioufly ; fo that fome- 

 imes the one prevails, and fometimes the other. 



It has been a point of inquiry to find the pofition of the 

 udder, in which the advantage (hould be the greateft. 

 A. Renau, in his famous theory of the working of fhips, 

 as found, that the beft fituation of the rudder is, when 

 : makes an angle of fifty-five degrees with the keel. See 



lUDDER. 



If, now, a wind-mill, expofed direftly to the wind, (hould 

 lave its four fails perpendicular to the common axis in which 

 hey are fitted, they would receive the wind perpen- 

 iicularly ; and it is vifible that impulfe would only 

 end to overturn them. There is a necelfity, therefore, 

 have them oblique to the common axis, that they may 

 eceive the wind obliquely. 



For the greater eafe, let us only confider one vertical fail. 

 rhc oblique impulfe of the wind on this fail is reducible 

 a perpendicular impulfe ; and that direftion, as the 

 ail cannot abfolutely keep to it, is compounded of two ; 

 ne of which tends to make it turn on its axis, and the 

 thcr to fall backwards. But it is only the firft of thefe 

 lireftions that can be obeyed. Of confequence, the whole 

 mpulfe of the wind on the fail has no other effeft but to 

 aake it turn from right to left, or from left to right, 

 s its acute angle turns this way or that. And the llruc- 

 ure of the machine is fo well contrived, that the three 

 ither fails are determined, from the fame caufes, to move 

 he fame way. 



The obliquity of the fails, with regard to their axis, has 

 irecifely the fame advantage and difadvantage with the 

 ibliquity of the rudder to the keel. And M. Parent, 

 eeking, by the new analyfis, the moft advantageous fitu- 

 tion of the fails on the axis, finds it precifely the fame angle 

 f fifty-five degrees. 



For the farther illuftration of this point, let A B ( Plate 1 1 . 

 Vtnd-MlU, Jig. 7. ) be the axis of the mill, C D a fail, and its 

 ngle of obhquity (wz. that which it makes with the axis) 

 e E C G ; then if G C be the force of the wind in the di- 

 |e£l; pofition of the fail, G E (the fine of the angle of inci- 

 !ence GC E) will be the force of the wind in its oblique 

 j ofition ; but the force of G E is refolvable into two 



Vol. XXXVIII. 



others, E F and G F ; of which the latter, being parallel to 

 the axis, avails nothing in turning the fails about it ; but the 

 other, E F, being perpendicular to it, is wholly fpeiit in 

 compelling the fail to turn round. The force of the wind 

 on the fail will be as the fquare of the fine of incidence, or 

 as GE'; and if the area of the fail, and the velocity 

 of the wind, be fuppofed conltant, the force of the wind 

 in the direft pofition will be to that in the obhque 

 one, as G C to G E ' ; but when G E is the whole force, 

 that part which turns the fail is reprefented by E F ; 



andGE : EF ( :: GC : CE) :: GE' 



CE X GE' 

 GC 



= to the force which turns the fail, when the whole 



force is reprefented by G E ' . This expreffion 



CE X GE' 

 GC 



begins from nothing, when the angle of incidence begins 

 to be oblique, and increafes with the obhquity of the 

 faid angle to a certain number of degrees ; becaufe 

 that part of the force which is parallel to the axis, be- 

 comes lefs in proportion to that which is perpendicular 

 to it ; but after it has pafled this limit, it again de- 

 creafes, and becomes nothing, when the angle of incidence 

 vanifhes. There is, therefore, one certain pofition of the 

 fail, in which the force of the wind upon it is a maxi- 

 mum. In order to find this, put radius G C = a, E C = 

 X ; and we have CE.^ z= aa — x x, and confequently the 



f CExGE^ 



torce — 



-, which muft be a maxi- 



GC - a 



therefore its fluxion aax— ■3,x''x=^o: whence 



and fo X 



v/?= 



(in logarithms) 



20.000000— 0.477121 



= 9.761439, which is the loga- 



rithmic fine of the angle 35° i6'=CGE; and there- 

 fore the angle E C G = 54° 44', when the force of the 

 wind is a maximum. Thus, alfo, if / m [Jig. 7. ) parallel 

 to the axis Q M, be equal to a, and reprefent the whole 

 force of the wind on the fail ; this force is reduced to / n, 

 and this again to n 0, which afts perpendicularly to the 

 axis, and turns the fail. This force, putting m n = u. 



is exprefled by 



-, and thus, as before, when it 



IS a maximum 



.«=v/^«=Vi= 



3 ^3 



Martin's Phil. Brit, vol 



and the angle 



I. p. 22c, 



Imn =■ 54° 44'. 

 vol. li. p. 212. 



This angle, however, is only that which gives the wind 

 the greateft force to put the fail in motion, but not the 

 angle which gives the force of the wind a maximum upon 

 the fail when in motion : for when the fail has a certain 

 degree of motion, it yields to the wind ; and then that 

 angle muft be increafed, to give the wind its full effed. 

 Mr. Maclaurin, in his Fluxions, vol. ii. p. 734. has fhewn 

 how to determine this angle. 



It may be obferved, that the increafe of this angle ftiould 



be different, according to the different velocities from 



the axis to the extremity of the vane or fail. At the axis 



it fliould be C4° 44', and thence continually increafe, givinj 



3R the 



