ON THE GENERAL THEORY OF THE STEAM ENGINE. 



1/9 



traversed by the connecting rod of that crank which takes the lead in the direction of rota- 

 tion, and y the corresponding distance traversed by the connecting rod of the other crank 

 which follows it. Then, substituting the value of R, &c., in (17), we have, 



= v W 2 + ^ ( + v - ^ ) 



Now, referring to the figure on page 2, we shall suppose the advancing crank to proceed 

 from the point E with the angular velocity (<>), and to pass over to the four successive 

 positions D, D", D, D', equidistant from the diameter E H. The other crank will proceed 



from the point F, and always be 90 or behind the former. Let ft denote the arc E D or 



H D", and assume, 



h = r - 



- p 2 sin. 2 /? 



k = r - V r* - p* cos. 2 /3 



m = r - 



---- (20). 



Then we shall have, for the point D, x = p (1 cos. ft) + h, 



y = p + m p (I sin. ft) k; for the point D", x = p (1 + cos. ft) + h, 

 y = p + m + p(l sin. ft) + k; for the point D, x = 4 p p (I + cos. ft) h, 

 y = p + m + p(l + sin. ft) + k; and for the point D', x = 4 p p(l cos. ft) h, 

 y = p + m + 4:p p (I + sin. ft) k. Hence the sum of the distances x + y takes 



the following values, 



at D, x 4- y = p (1 cos. ft + sin. ft) + m + h k, 



D", ' = p (3 + cos. ft sin. ft) + m + h + k, 



D, = p (5 - cos. ft + sin. ft) + m - h + k, 



D', = p (7 + cos. ft sin. ft) + m h k. 



Also, for these four points the angles described are respectively ft, IT ft, tr + ft, and 2 IT ft, 

 which must likewise be severally substituted for a in (19). Performing these substitutions, 

 and making, 



A = p(i/8-l+ cos. ft - sin. ft } (21), 



* IT ' 



we find, for the four respective points just enumerated, 



= ,/ (a,) 2 + --(m- 



JV1 



*) 



= V (a,) 2 





(22). 



These expressions may be considerably simplified by disregarding the powers of the small 

 variations above the first. In the first place we immediately obtain, 



