Aug. i6, 191S Tensile Strength and Elasticity of Wool 389 



Integrating with respect to u, we have — = —^~~ — g — %^- Substi- 



dk -yjz -yjr^ + r^ 



.4769)^ 

 tuting , , = s and putting the equation into the integral form, 



we shall have q = - I e~"'du. Now, tables are made for the integral 



X 



^f 



e '^dt when t is a constant. Now, /? = Wj — w^ + h; r^ and ^2 are known, 



so q can be taken from this table. Returning to p, we have p = — q. 



The value of q evidently depends upon the ratio of — fe to -ylr^ + r^. 

 Taking these two values as the ordinate and the abscissa of a point, it is 

 evident that all points on a straight line passing through the origin of 

 coordinates would give the same value p. Given p, we may determine 

 the slope of the line corresponding to the given value of p. On a sheet 

 of cross-section paper (fig. 3) lines have been drawn corresponding to 

 certain values of p. 



As an example of the use of this diagram, observations on two sets of 

 wool gave as their tensile strengths the following results: (a) ioi,i8o±. 

 2,219 and (6) 75. 293 ± 1.573 dgm. per sq. mm. Wj-Wa^ 25,887 

 ■\lr^ + r.^= (2,219)2+ (1,573)2 = 2,719.05. Taking each of the smallest 

 divisions as 200, we see that the chances are more than 9,999 out of 10,000 

 that the fibers in group a are the stronger. Now, taking h successively 

 as 5,000, 10,000, 15,000, 20,000, and 25,000, we shall have the quantity 

 m^ — m^ — h taking successively the values 20,887, 15,887, 10,887, 5.887, 

 and 887. Taking each of the smallest divisions as 100, we see that the 

 chances are more than 9,999 out of 10,000 that group a is at least 10,000 

 dgm. per sq. mm. stronger than b. The chances are practically 995 

 out of 1,000 that group a is at least 15,000 dgm. per sq. mm. stronger 

 than h. The chances are 92 out of 100 that group a is at least 20,000 

 dgm. per sq. mm. stronger than h and 6 out of 10 that it is at least 25,000 

 dgm. per sq. mm. stronger than h. Reading these facts in another way, 

 we may state with the probability of being correct 60 per cent of the 

 time that the fibers of group a are at least 25,000 dgm. per sq. mm. 

 stronger than those of group h; 70 per cent of the time that group a 

 is at least 23,700 dgm. per sq. mm. stronger than group h\ 75 per cent 

 of the time that group a is at least 23,150 dgm. per sq. mm. stronger than 

 group 6; 80 per cent of the time that group a is at least 22,500 dgm. 

 per sq. mm. stronger than group b; 85 per cent of the time that group a 

 is at least 21,750 dgm. per sq. mm. stronger than group 6; 90 per cent 

 of the time that group a is at least 20,700 dgm. per sq. mm. stronger than 



