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VIII. The Practical Importance of the Confluent Hypergeo- 

 metric Function. Bq H. A. Webb, ALA., and John 

 E. Airey, AT. A., D.Sc* 



[Plate VI.] 

 § 1. Introduction. 



IT is well-known that many physical and engineering 

 problems depend for solution on differential equations 

 of the type 



§ +/W. £+*<*)• y-o, . . . (i> 



where /( x) and <£(V) are given functions of x. For example, 

 the investigation of the periods of lateral vibration of a 

 flexible non-uniform rope or chain f> or the periods of 

 vibration of a circular disk J, leads to an equation of this 

 type. Again, the whirling speed of a non-cylindrical shaft, 

 or the period of lateral vibration of a non-cylindrical bar, 

 such as an air-screw blade, can be found, with two-fio-ure 

 accuracy, by the solution of such an equation§ ; and in fact 

 many vibration problems in various branches of physics lead 

 to such equations. To take another illustration, the crippling 

 end-load of a tapered aeroplane strut, whatever law of taper 

 is adopted, could be found if we could solve equation (1); 

 other problems of elastic instability lead to equations of this 

 type, and may be brought into prominence in aeronautics by 

 the urgency of saving weight. 



In structures, such as aeroplanes or bridges, the liability 

 to secondary failure (•". e. elastic instability) must be foreseen 

 and estimated, as well as the liability to primary, or stress, 

 failure. In running machinery it is important that the 

 period of free vibrations shall be well above, or below, the 

 given running speed, to avoid resonance; in instruments for 

 producing sound, on the other hand, it is required that the 

 period of free vibrations shall have a given value, to secure 

 resonance. 



In any of these cases, the problem presents itself to the 

 designer somewhat as follows. The main outlines of the 



* Communicated by the Authors. 



\ Airey, " The Oscillations of Chains," Phil. Mag. June 1911. 

 X Airey, " The Vibrations of Circular Plates," Proc. Phys. Soc. April 

 1911. 



§ Webb, " The Whirling of Shafts," Engineering, November 1917. 



Phil. Mag. S. 6. Vol. 36. No. 211. July 1918. K 



