ON THE GENERAL THEORY OF THE STEAM ENGINE. 177 



which are here arranged respectively in the order of their magnitudes, that at D being the 

 least, and that at D" the greatest. 



These expressions may be put in a more convenient form by neglecting the higher powers 

 of the small variations. We shall then have, 



= () - J^_ (A + h}, 

 P (A - h), 



MH 

 ' = () + ^JL (A - A), 



-" = () + JT-T ( A + *) 



But for h we may now substitute ^ sin. 2 /3 ; therefore, 



= (,) - .J^_ (A + P- sin. 2 /3 ) 

 M (a,) V 2r I 



p / * 



M (>) 



_ sin. 2 /3 ) 

 9.r ' 



L_ f A - P 



- sin. 2 /3 ) 

 2 r / 



( A 



sin/ 



(15); 



M() V 2r 



in which the values of A in parts of p are by (13) found to be as follows : 



ft A 13 A ft A, 



0-0000 30+ -1994 60+ '1667 



5 + -0518 35 '2080 65 -1448 



10 -0959 40 -2105 70 -1198 



15 -1326 45 '2071 75 -0922 



20 -1619 50 '1984 80 '0625 



25 -1841 55 '1847 85 + '0316 



30 + -1994 60 + -1667 90 0-0000 



From these last expressions we conclude that the velocity M which occurs at E and H is the 

 mean velocity, and that the greatest and least velocities occur in the quadrants of D" and D, 



o 2 2 p 



at the points where A + sin. 2 /3, or ft + cos. ft + -. - sin. 2 /8, attains its maximum 



2 r TT 2 r 



value. If we assume these points to be in the middle of the respective quadrants, so that 

 /3 = 45, which will be very nearly the case with all the proportions observed in practice, and 

 cannot sensibly affect the accuracy of the results, the greatest and least velocities will be 



M + , , x ( - + 0-207 ) and the deviation from the mean value will be, 

 " M M * r 



fL ( -- +0-207). 



M M v 4 r ' 



