36 Mr. J. H. C. Searle on the Effect of 



the corresponding normal functions for the various cases 

 which may arise. The complexity introduced into the normal 

 functions is not very great, and it is some mental satisfaction 

 to see how the correction suggested by the use of unmodified 

 normal functions actually arises as an approximation to the 

 accurate equations. The sequence of my paper will be 

 (1) to obtain full equations for the notes in the six possible 

 combinations of terminal conditions, (2) to solve these nume- 

 rically and table the values of the fundamental note and the 

 overtones. Lastly (3) I shall discuss the form of the suitable 

 normal functions in these cases,, and the fundamental integral 

 on which not only solutions in these functions depend, but 

 which determines also the influence of a generalized force of 

 corresponding type on the vibrations of the bar. 



II. 1. Equation of Motic 





OL 



Let OQ be a straight bar of uniform cross-section and 

 material. 



Let I = length of bar. 



v = displacement transversely of an element distant x 



from 0. 

 m = mass per unit length of bar. 

 A = area of cross-section. 

 k = swing radius of cross-section about a line through 



centroid perpendicular to plane of bending. 

 E = Young's Modulus for the material of the bar. 

 S = shear across bar \ at [nt 



M = bending moment J c - " 

 Y = transverse body force. 

 Then 



and 



where 



mv=^~+mY (1) 



d z v dM ,,. 



<7^. = S+ ^ ( 2 > 



g=ian.a=*,q.p (3) 



