742 Prof. J. E. Ives on an Approximate Theory of an 



this series. In these cases, closer values are obtained by- 

 interpolation from tables of the J , J 1? Y and Y x functions. 



The following table gives the roots of equation (2) for 

 different values of A ; the earlier roots were found by inter- 

 polation, the higher roots from the general expression given 

 above. 



Table of roots of equation (2). 

 J (\*) J 2 (z] 



Y Q (\z) Y 2 (z) 



, \>1. 



Value of X. 



First rout. 



Second. 



Third. 



Fourth. 



s/Tl 



6380 



65-264 



129-587 



194-116 



Vi-2 



4-540 



33-536 



66-145 



98-956 



Vi-6 



2-920 



14-581 



28-268 



42-144 



ViTo 



2-114 



8-147 



15-441 



22-908 



1-5 



1-905 



6-852 



12-870 



19-055 



Vao 



1-534 



4-837 



8-882 



13-078 



2-0 



1-278 



3-659 



6-594 



9-626 



30 



0-813 



2 017 



3-416 



4-906 



40 



0-604 



1-431 



2-352 



3-328 



5-0 



0-482 



1-122 



1-794 



2-586 



— 



LXXXVIII. An Approximate Theory of an Elastic String 

 vibrating, in its fundamental mode, in a Viscous Medium. 

 By James E. Ives, Ph.D., Associate Professor of Physics 

 in the University of Cincinnati *. 



IN the theory of the elastic string, usually given, the effect 

 of the internal and external friction is assumed to be so 

 small that it can be neglected. In certain cases, however, 

 this is not permissible. For instance, if the string vibrates 

 in a viscous medium the external friction can no longer be 

 disregarded. 



Since the displacement and velocity of any point on the 

 string vary from point to point along it, the system is really 

 one having an infinite number of degrees of freedom and is 

 difficult to treat. An approximate solution may, however, be 

 obtained by making use of its mean velocity, and regarding 

 it as a system having only one degree of freedom. To do 

 this, we must know the transverse displacement, q x , of the 



* Communicated by the Author, 



