I I'-' ] 



II. 



EXTENSIONS OF FOUEIEE'S AND THE BESSEL-FOUEIER 



THEOREMS. 



Emekdations and Additions to a Pkevious Paper. 



By PEOFESSOE WILLIAM M'FADDEN OEE, M.A., F.E.S. 



EeaJ Februauv 13. Published May 13, 1911. 



Contents. 



1. Minor Corrections. 



2. Defects of the former Expansions, especially in regard to Differentiability 

 and Uniqueness. 



.3. The Expansion in Trigonometrical Series modified so as to refer to Two 

 Arbitrary Functions. 



4. The Expansions still Not Unique : Terms which may be added. 



5-8. Application to Transverse Vibrations of Elastic String whose ends 

 are attached to a System of Particles with Massless Elastic Connections. 



5. Equations to be satisfied : the Actual Solution. 



6. This Solution satisfies the given Initial Conditions. 



7. This Solution satisfies the Terminal Equations. 



8. This Solution satisfies the Equation of Motion of the String. 



9. Application to a Problem in Heat-Conduction. 



10. Eeference to another Class of Physical Problems to which Similar 

 Expansions apply. 



11. Eeference to the corresponding Modification of the Trigonometrical 

 Integral Expansion. 



12. The corresponding Modification of the Expansions in Bessel Functions, 

 and application to a Physical Problem. 



13. Expansions Apphcable to Vibrations of Elastic String, itself subject 

 to Viscous Forces. 



14. A Circular or Annular Membrane, itself subject to Viscous Forces. 



15. The above Solutions agree with those found by the aid of the 

 characteristic Property of Normal Functions. 



