180 THE SHORTER SCIENTIFIC PAPERS 



due to spinning, one may substitute s/sin^ for (s) in 

 equation (1) and the average length (I2) is indicated by 

 equation (2). 



l2=2 (s/sin 6) n/e (2) 



If, however, the axial fibers are under tension, a dif- 

 ferent mathematical treatment is needed, since the lengths 

 of the assumed continuous fibers will vary from the 

 lengths of the axial fibers (s) to those of the peripheral 

 fibers (s/sin ^) , the distribution about the axis corres- 

 ponding to the square of the radius. Using (R) for the 

 radius of the segment and (r) for the varying radii, the 

 total length of all helices (Sh) is exhibited by equation 



(3). 



2li=(sn/7rR3 tan d) C^lirr^r'^ + R- tan^ 6 dr (3) 



J o 



and the average length (I3) is shown in equation (4). 



13=4 sn • tan2 6 (csc^ ^—1) /3e (4) 



The differences in the values obtained by (1), (2) and 

 (4) are relatively small and experiments with fibers of 

 known length are closely in agreement with the theoretical 

 values. Any factor for average fiber length will need a 

 modifier, probably of an exponential nature, determined 

 in connection with subsequent experimental work, since 

 the increase in strength of yarns will not continue to be 

 proportionate to the increase in fiber length. 



The development of standards for materials along the 

 lines suggested, presenting something more than arbitrary 

 objective tests, is decidedly desirable under our present 

 economic system. Such studies are now in progress. 



The writer wishes to acknowledge his indebtedness to 

 Dr. R. B. Allen, professor of mathematics at Kenyon 

 College, and to Mr. Graham Walton, instructor in engin- 

 eering at the University of Wisconsin, for assistance in 

 connection with equation ( 3 ) . 



