OP ARTS AND SCIENCES. 51 



IV. 



NOTE ON AN APPROXIMATE TRIGONOMETRIC EX- 

 PRESSION FOR THE FLUCTUATIONS OF STEAM 

 TEMPERATURE IN AN ENGINE CYLINDER. 



By Edwin H. Hall. 



Presented January 11, 1893. 



During one of the long interruptions of an experimental inves- 

 tigation of cylinder condensation in steam-engines,* I was moved 

 to attack the problem mathematically, making use of the well 

 known methods of treating heat conduction, which are set forth to 

 advantage in Riemann's Partielle Differentialgleichungen. I car- 

 ried the process far enough to see that the waves of heat produced 

 at the inner surface of the cylinder wall, by the fluctuations of 

 steam temperature during each stroke of the piston, would pene- 

 trate to a considerable depth; but the labor of a complete inves- 

 tigation of the problem by this method seemed likely to be much 

 greater than that required by the experimental method, while the 

 results would necessarily be open to considerable doubt, owing to 

 the assumption that one must make in estimating the temperature 

 of the inner surface of the cylinder wall at any point of the stroke. 

 I therefore returned to my experiments, convinced that experi- 

 ment alone could give the results for which I was striving. Never- 

 theless, it may be of interest to those concerned with problems of 

 heat conduction, theoretical or practical, to see the result of the 

 first mathematical steps, the derivation of an approximate trigono- 

 metric expression for the periodic changes of temperature suffered 

 by the steam during each complete stroke of the piston. 



The indicator diagram, Figure 1, gave the pressure, and so the 

 temperature, of the steam at every point of the stroke. Assuming 

 the engine piston to have simple harmonic motion, that is, neglect- 

 ing the effect of the vertical movement of the crank pin, I plotted a 



* See ante, page 37. 



