826 Transactions of the American Institute. 



8 = the space passed over by the piston. 



r = the length of the crank. 



t = time ; v = velocity ; f = accelerating force. 



F = constant value of f at maximum. 



T = constant time of revolution. 



Y = constant angular velocity of revolution. 



Then V-**-?Z. **- - J&L 

 men v - ^ - -^ ^ a _ T „ ^ 



<?5 = y* 6? ( v. s. y) = r 6? ( — cos. <p ) = r sin ?> ^. 



tfe _ ?• sin <p d<p dv r d (sin <p d<p) r cos.?' dy 1 



V ~ ~dJ~ di . df d£* ~" cP 



Substituting for — vf— ~jr = f p^ • ^ C03 - f 



But ~^r = ^ • • ^ = V* - r °° 8, * 



"When cos. <p = 1 or — 1, / = + v ^ r = -f- V 3 ?* = centrifugal 



force of a unit mass of matter revolving in a circle of which r is the 

 radius, the time of revolution being T. The accelerating force f, 

 therefore, varies as cos. y, and is maximum when cos. <p is maximum. 

 Hence: 



F 



- ^ and -*L - V*? r - 



T 2 ',""- ^ — 32.I66 T 2 ~ 8.0416 T 2 

 which expresses the ratio of F to gravity. 



In order to compare this force with the constant force which would 

 generate the same velocity in the same time, we observe that, by 

 hypothesis, this force acts only during a quarter revolution of the 

 crank, or a half stroke of the piston ; and, hence, that the maximum 

 velocity is necessarily equal to the uniform velocity of the extremity 

 of the crank ; or, in other words, since 



r since d<p -, dip 2tt . . 2~r 



v " — i~; and to = "T.; ■ • V = S11 ^-T-. 

 which is maximum when sin^ is maximum, or when <p = 90°. 

 Thence, putting Y 1 for this maximum velocity. 



• y - J^L 



' 1 rp 



This velocity is generated on the time ^T. Supposing it gene- 

 rated by a constant force, this force would be represented by the 

 velocity it is capable of generating in one second ; or putting F x to 

 stand for this constant force. 



r i - £T ~ +T 2 "~ *"^~> 



