Record. xlv 



There are therefore two positions of the plane for which P= R, or 

 for which 



Sin a + /cos a = 1 



This equation when solved as a quadratic gives 

 1 4- f 2 



Hence 



If the force P is decreased to some value P t so that the block slides 

 uniformly down the plane, then 



Pi = R' —fR" = P(Sina —/cosa) (2) 



This force Pi is the force which horses must apply to a wagon in 

 descending a hill, if they are to maintain uniform motion. 

 Equations (1) and (2) may be written 



y = sin a -\- /cos «. 



When the final term has the + sign, y = P, and for the — sign, 

 y = Pi. The two curves which represent the values of y for direct and 

 reversed motion on the plane, are symmetrical in y with respect to the 

 sine-curve which represents both direct and reversed motion on the 

 frictionless plane. As / becomes smaller the two curves approach the 

 sine-curve. 



The work done in raising the mass m through any height heing W , 

 the work required to slide the mass up any plane having any angle of 

 inclination through a vertical height h a is 



W= W Q (1 4- /cot a) 



This work increases as a decreases. It is W when a = 90°, and it is 

 2 W a when tan a = f, or the angle of the plane is the angle of friction. As 

 a diminishes to zero, the value of W increases to an infinite value. 



Professor Winthrop H. Chenery addressed the Acad- 

 emy on "The Problem of Organization; a Criticism of 

 Theodor Ostwald's Der energetische Imperativ." 



Professor C. A. Waldo described "A New Application 

 of Electricity to the Production of Musical Tones." 



