103 



Hard cast-irou shoe. 

 Heavy pressure. 



Speed increasing from 10 to 6.5 miles per hour. 

 Coefficient of friction decreasing from 25 per cent, to 12 

 per cent. 

 3. The coefficient of friction of cast-iron brake shoes is practically 

 constant with variations in temperature of shoe and wheel within the 

 limits of the experiments. 



Diamond Fluorescence. 



[Abitract.] 



By Arthur L. Foley. 



A year ago I presented to the Academy an account of an experiment 

 with a diamond and a photographic drj^ plate (Proceedings of Academy. 

 1899, p. 94). Later experiments have confirmed the theory presented. 

 It has been found that a low temperature is favorable to the success of 

 the experiment. 



A Theorem in the Theory of Numbers. 

 By Jacob Westlund. 



Let n be any prime number and let 



Sk = li' + 2M 31^+ + (n — 1 k. 



Then 



Sk = 0, mod n, when k ± 0, mod (n — 1) and Sk = — 1. mod n, when 

 k~0, mod (n-1). 



Proof. Consider the congruence. 



x"-i— l = (x — 1) (x — 2) (x — n — 1), mod n. 



This congruence is evidently satisfied by the n — 1 incongruent numbers. 

 1,2,3, (n-1). 



