Polytechnic Association. 841 



•when <p = 45°. Also, that the work performed through the crank 

 during the second half-stroke, by the gradual transformation of 

 this living force of the reciprocating mass into work, is expressed by 



the analogous formula, ~ — £g i* cos - 2 <?• And this is equally divi- 



ded at the point of the stroke at which cos. 2 p = \ ; which is once 

 more true when 9 = 90° + 45° = 135°. 



Furthermore, when steam is worked expansively, say with a cut-off 

 of one-eighth, the work done is expressed by the formula, 



P *( 1+M 1), 



for the first half stroke ; and by the formula, 



»ji+hi*]_M(i+M.jj; 



for the second half stroke. 



If we put x to stand for the distance over which the piston must 

 pass in performing one-half the work of the half stroke, we shall 

 have the two equations, 



2P& f 1+h l| 1 =Vk f 1+h 1 -j ] , for the first half stroke ; and 

 2PA f l+hl-M =P& f 1 + hl^l —Pk fl+lil-J'] , for the 



second. Whence, in the former case, l + hlyT-^^^O. 

 If r=l, and h=^, h 1 4ar= — 1 ; and h 1 2x= — A- ; or 

 hli= + i =0.5. 



And a?=0. 30327, which is the natural versed sine of the angle 

 45° 50'. 



The total work of the steam is thus divided, in the first half stroke, 

 very nearly at the middle point of the quadrant, and the amount of 

 this work which the inertia of the piston absorbs, is divided exactly 

 at that point. "Whence the portions of this work which are done in 

 the successive octants are very nearly the same. 



In the second case we have immediately, 



h 1 -p-^h I-75- or a? = 2r 2 . 



Whence x = r -\/¥~ ; or putting r =1, x = 1.4142, which is the 

 catural versed-sine of 114° 30 ; ; say 115°. 



The work done by the steam in the second quadrant is equally 

 divided, therefore, at a point not one-third advanced from the begin- 

 ning, but that which is added by the living force of the reciprocating 

 mass, which is the greater of the two, is equally divided, in this as in 



