176 



APPENDIX. 



distance through which it has acted. In a semi -revolution, the power developed by P is 

 hence P x 2 p ; that expended by R would similarly be R x TT p ; and, according to the equa- 

 tion (6), it appears that the expenditure would be the same in both cases, and that no loss 

 is sustained through the obliquity of the action of the crank, except that which arises from 

 the additional friction caused by the stress on the shaft. 



Substitute the value of R by (6) in (5), and, for permanent speed, the angular velocity of 

 the shaft of the engine at any period of the stroke is, 



(7); 



and for any position D from E to H, if /8 denote the angle E O D, this becomes, 



- cos. /3 ) - f 



- p 2 sin. 2 /3 ) ...... (8). 



For a point D in the returning half stroke from H to E, if /3 denote the angle H O D, the 

 value of A C' is obtained by substituting TT ft for a in the value of x expressed by equation 

 (1), and is therefore, 



A C = p (1 + cos. /3) + (r - 



(9). 



-/> 2 sin. 2 /3); 



and, if we deduct this from the whole distance A B = 2 p, we get the expression for B C or 

 x', viz. 



af = p(l- cos. /3) - (r - A 

 Therefore by equation (7) the velocity at D is, 



/TV 



-p 3 sin. 2 /3) 



M 



( c + p (1 cos. /3) ft (r 



\ IT 



- 2 sn. 2 



) 



(10). 



The velocities w <*>" at D' D" will now be determined by substituting TT /3 for ft in equations 

 (10) and (8) ; thus we find, 



/T 



r 



JVL 



~ cos - 



*JL ft -fa- 

 ir 



- 2 



p 2 sin. 2 /3) (H), 



o>" = V 4r- ( c ~ P ( l ~ cos - #) + & + ( r ~ V ^ - P 2 sin. 2 /3) ) . , . (12). 



M v 7T / 



To put these in a more simple form for comparison, let (<) denote the velocity at E, as 

 expressed by equation (a), and assume, 



2 



7T 



h = r - 



A = p ( - ft - 1 + cos. 



\ 7T 



(13) 



r 3 p 2 sin. 2 /9 

 Then, the angular velocities at any four corresponding points D D D' D" are, 



= V W 2 - 



(A + h) 



, = ()+ 



(A-*) 



(14); 



<o" = V (a,) 2 + 



(A + A) 



