I^ On the "whirUng Eddies of demanding Water, 



have nearly ©=-;< = >•*; and the centrifugal force of the particle D will be = -j^. When we 



attentively obferve the particles which revolve at the furface of the funnel, at M N, we fee 

 that the effeiS which really takes place in nature, is nearly /=r^ Since, therefore, the cen- 

 trifugal force in approaching the centre C increafes as — , it will become equal to forming 



an equilibrium againft the upper preflure S D, which produces the centripetal force D C. 

 A cavity, K R T H P v, will therefore be formed, round which the whirling fluid will fupport 

 itfelf by the centrifugal force of its rotation. 



Let D Q_P R reprefent a circular fluid zone, the particles of which turn round the cavity 

 R P, according to the law here indicated. Let the gravity of a fluid particle be =.g ; C R 

 ■=a; R D=*; D X = z; X Z=</z; and the velocity of the particle D = v. If the centri- 

 fugal force of the particle D were equal to its gravity, its velocity, by the thorems of Huy- 



o-ens, would be equal to that of a body falling by gravity alone, through the fpace — — — ♦ 



And fince an heavy body falls in one fecond through the fpace of i8i inches = S, the velocity 

 of the particle D on the fame fuppofition would be = V (2 S (« + ^) ). The centrifugal force 



in the circle is as -o"-, the centrifugal force of D will therefore really be = 7" ^ ■' And 



III — "yV 



fince the centrifugal force is = --; taking-- — — — - : - — — -= ^ ^ , , ,. : a fourth 



° r^ {a-ifby («+* — zj' 2 b \a-\-b) 



term, we (hall have the centrifugal force of the element of D X in X= ^— ; — ; 



20 («+* — zp 



Bud that of the filament D X = AH — ^-f-^-, — -Tx' When z=o the integral is =0; whence 



4S (a + b — z)* 



A —JH—L.. Taking: zzzL the centrifugal force of the filament D R will be = — 2__. 

 4 S o ' & 4 a'' S 



(a a + b). The quantity i j' is the gravity itfelf of the filament D R. The gravity of this 



filament is therefore to its centrifugal force =t>* (2 a+b) : 4 «* S. 



When the fluid zone, D R P Q, is nearer the aperture E F, the preflure S D increafes ; 



whence the centrifugal force of the zone muft alfo be increafed by diminifhing the radius of 



the cavity R C : hence we may determine the nature of the curve which forms the perpen- 



tlicular feftion of the cavity K R T, For greater fimplicity, let us fuppofe that the fides of the 



veflel have the fame form M D as that of the cavity itfelf, fo that D R = ^ may be conftant. 



Let A C = Ar; and C Rz=y. Let us fubftitutey inftead of a in the preceding formula. And 



fince the gravity of the filament D R, is to the gravity of the filament S D=^ : at,, we fhall 



have by compofition of ratios, the centrifugal force of the filament D R, to the preflAire S D = 



i» V (2 y + b) ; 4 x/ S. Thefe quai>tities muft be equal> in order to afford an equilibrium. 



We have therefore* v^— — s — =afor the equation of the curve K R T. This 



,26 -4 1> 



is the fixty-fourth fpectes in the enumeration of lines at the third order, by Sir Ifaac Newton. 



Its convexity is turned towards the axis; it lias two a fymptotes, one of which is the axis 



AY, and the other is in M N, fuppofing the two points M N to be infinitely diftant. 



If 



