28 AN ACCOUNT OF THE EARLY MATHEMATICAL AND 



and 1g = 32J, the force of gravity. Then, per mechanics, 



the weight of w in direction AC = ~^, 



or in opposition to the weight W : — 



consequently W T ^ = the weight 



acting in direction of the thread; the 



negative sign taking place when C is 



below A and the affirmative when 



above. Now it is obvious that if W do 



not descend, but continue at rest, the 



centrifugal motive force of w, divided 



by gravity, must be equal to this 



weight. Or ^ = W T ^; whence 



° 2 g V r 



v = |-^ (Wr =F 2^ x)\^, for the required velocity of w, or 



of the point b of the plane when the weight W continues at 

 rest. 



When the plane is vertical below A, x = r, and v = 

 J -JLI ( W — k;^ l.2j being then a minimum ; if a; = c, or the 



plane horizontal v = (^l^-^ — ^2. jf ^ ^^ r, and the plane be 

 above A, t; = /H-^(W + w)Y> being then a maximum: — 

 lastly, if W = 2^, and the plane vertical below A, v = o, or 

 the weights being equal, balance each other. If w be made 

 to ascend with a force which is to gravity as w : 1 ; then we 



shall have - — - = W ^ 1^ + ^ ^j ^^^ consequently v = 

 |-^(1 + w) W ;• T 2^ a;| ; and when W is made to descend 



with a like force, n must be negative ; but not so great as to 

 make the quantity under the vinculum negative." 



We have now arrived at the period when Mr. Dalton 

 discontinued his correspondence to the mathematical periodi- 

 cals. He had recently removed to Manchester, and the 

 conscientious discharge of his duties, as professor of mathe- 

 matics and natural philosophy, left him but few opportunities 

 of indulging his taste for pure mathematics. The circum- 



