Record. xlv 



There are therefore two positions of the plane for which p ^ R, or 

 for which 



Sin a -\- fcoa a = 1 



This equation when solved as a quadratic gives 



e; 1 + r- 



Hence 



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

 uniformly down the plane, then 



Pi^ R' —fR" = R^'riin a— f cos a) (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 u -|- /cos o-. 



When the final term has the + sign, ;/ — 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 ?/ 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 being Wq, 

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

 inclination through a vertical height h^ is 



TF= Wo a +f cot a) 



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

 2 Wo 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. Cheuery addressed tlie Acad- 

 emy on "The Problem of Organization; a Criticism of 

 Theodor Ostwald's Der energetisclie Imperativ." 



Professor C, A. Waldo described ''A New Application 

 of Electricity to the Production of Musical Tones." 



