Polytechnic Association. 827 



From this we obtain 



F _ (2tt)V _ 2tt _ 1 

 F x " 4(27rr) ~~ 4 ~~ V 

 Al li - 4(2?rr) _ 8»r _ ^r 



9 ~ 9 P ~ 32.166 r P _ 4.0208 T 2 . 

 If we take, for r and T, the values assumed by Mr. Porter in 

 the Engineering and Mining Journal, viz., r = 1.25 ft. and T = 



■ sec., we shall obtain, by substitution, 



F = 205.031, and F x = 130.527 ; whence ^- = 1.57078 = £ * as above. 



This force ~F ly however, is a greater force than would be required to 

 generate the velocity, Y l5 in acting constantly through the space 

 represented by r, if time is left out of the question ; for, remember- 

 ing that the spaces passed through under the influence of constant 

 forces are governed by the law, 



S = \ft? 

 and applying this to the case in hand in which the time to be con- 

 sidered is ^T, we have 



S = p\ QTf = *f- x ±T = frr, 



which is less than the radius r. 



"Where different velocities are generated by forces acting through 

 the same space, these forces are proportional to the squares of the 

 velocities generated. And the velocities generated by the same force 

 acting through different spaces, are as the square roots of the spaces. 



Thus, gravity will generate, in acting through the space r, the- 



velocity = g 1/ JL = g K/lS = \/Tvg 



v iff v ff 



Hence, the force F /7 which, in acting through the same space r } 



would generate the velocity 4p— will be found by the proportion, 



/ , x 2 /27zry (2tt)2 



(*f*rg) ' (r^-J ' ' ff ' fffrn. 



which is one-half the initial accelerating force F. 



Hence, the force required to give to the heavy -piston the requisite 

 initial acceleration is double that which, by acting constantly through 

 a space equal to the length of the crank, would impart to the piston 

 the same final velocity — a proposition which is, moreover, made 

 self-evident by an inspection of the diagram below. 



Since the accelerating force varies as the cosine of the arc of revolu- 

 tion ; and since, in the circle described by the crank, if the length of 



