178 APPENDIX. 



Again, by substituting the values of F, R, the force V - R, which at each position tends to 

 accelerate the motion, is found to be, 



TV T r. f 1 P COS. a \ 2 i 



F - R = P \ sin. a ( J + 1 ) I (16). 



I v v r* (r sm. 2 a TT J 



It hence appears that the motion is, 



accelerated T CQS> a ("greater 1 2 



>when sin. a I r . + 1 is < > than - or 0'63. 



retarded J ^ r ~ P sin - [less J "" 



Values of this expression are shown in the table given at page 230 of the present work. 



In the preceding investigation we have only considered the action of a single crank ; but it 

 will be found to apply, with a slight modification, when the rotary force of the shaft is main- 

 tained by the action of any number of cranks making given angles with each other. We have 

 only to substitute, in place of P', in the right-hand member of the equation of motion, the sum 

 of the forces F arising from the several cranks ; and, by following out the same process, the 

 equation (5) will become, 



in which 2 (P x) includes the same term for each crank. Let (<>) denote the value of this 

 angular velocity when x = o, a. = o, and the expression, for any other position, will hence be, 



(17). 



By attending to the nature of the integration, it is evident that this equation applies generally 

 from any assigned position throughout the entire period of each revolution of the shaft, if we 

 give to the symbols x a. the following signification, viz. 



x, the entire distance travelled over by the^ 



extremity C, for each crank ; I each being estimated from that position in 



a, the angle described by the revolution of j which the velocity was (<a). 

 the shaft ; 



Let us take the practical case, in which the revolution of the shaft is maintained by a pair 

 of equal engines, and in which the equal arms of the two cranks are placed at right angles, so 

 that each of them may be in full action when the other is " on the centre." If we suppose 

 one entire revolution to be performed, the distances x traversed by the extremity of each con- 

 necting rod will be 4 p, the angle a described will be 2 tr, and the velocity will become 



. = V (o,) 2 + ^ (4PP + 4PP-2RP-7T) = V ()" +^(4P-R7r). When the 



engines have attained their regular speed, this velocity must correspond with (), since precisely 

 the same motions must recur in successive revolutions ; we must hence have 4 P R TT = o, 



.-. R = 1 P ..... (18). 



7T 



2 

 This value is the same as (2 P) in which 2 P is now the moving power, and we observe 



IT 



that it corresponds with the expression (6) for the single crank. 



To find a more convenient expression for the velocity at any position, let x be the distance 



