Polytechnic Association. 825 



engine under consideration, that the time during which the effective 

 tangential force exerted upon the crank fails to act near the centers is 

 very small, not only absolutely (in consequence of high velocity), but 

 also relatively to the whole time of revolution ; and that, therefore, 

 the center passage imposes no great tax upon the regulators. This 

 matter being disposed of, we may proceed regularly with the problem 

 before us. This problem presents several questions which must be 

 separately considered. These are : 



First. Supposing the heavy piston to start from rest at the beginning 

 of its course, and the crank to be maintained in uniform angular 

 motion by some independent regulator, what must be the law of 

 force acting on the piston, so that it may complete the first half 

 stroke without exerting any strain upon the crank, in the way either 

 of acceleration or of retardation ? 



Second. Supposing the force accelerating the piston to cease to act 

 from mid stroke onward, according to what law will the living force 

 embodied in the piston be imparted to the crank, supposed still to 

 be maintained in uniform angular motion ? 



Third. Supposing the force on the piston to be constant through- 

 out the stroke, as in working a cylinder steam-engine without a cut- 

 off, according to what law will this force reach the crank ; and what 

 will be the relative amount of work done in the first and second 

 quadrants of the revolution 1 



Fourth. The same questions as proposed in the case last speci- 

 fied, with the additional supposition that a cut-off is used. 



Fifth. The point of cut-off, and the pressure of steam in the cylin- 

 der before cut off, being given, to determine the conditions which will 

 secure equality of work in the successive quadrants of revolution, 

 and the nearest approach to uniformity of action upon the crank ia 

 the direction of rotation. 



To proceed with the first ease. 



Taking, as before, ? to represent the arc of revolution measured 

 from zero at the line of the centers, the differences of the versed-sines 

 of <p for equal successive minute intervals of time will be proportional 

 to the velocities of the piston in such successive intervals. And the 

 differences of these differences will be proportional to the successive 

 accelerating forces required, in order that the uniformity of revolution 

 may be maintained. These conditions may be best expressed in the 

 notation of the differential calculus. 



Put therefore: 



f = arc of revolution, as above, to radius = 1, 



