for determining Frequencies of Lateral Vibration. 429 



have values differing by a finite quantity on opposite sides 

 of the support. It' therefore we are given that A is a 

 section of support, but no longer an end section of the 

 rod, and that Y vanishes at A, then this condition can be 

 realized by combining two solutions as before, and we obtain 



definite values of -^— and of: M wherewith to begin the portion 



of the curve which lies to the right of A. In our first 

 attempt at determining the curves for the next unsupported 



span we may assume that -=— does not undergo a change 



of value at A : if, then, at our second attempt we start from A , 



zero values for Y, 7 — , and M, but a finite slope — =— 

 ax d.v 



in fig. 3, and proceed for the span on the right of A by 

 the methods which have been described above, the 'curves 

 so obtained can be combined in any proportion with our 

 previously determined curves for the double span, and thus 

 we can satisfy the conditions at the next point of support. 

 It is not until we reach the other terminal section that 

 trial-and-error methods (in which 11 is varied) have to be 

 introduced. The labour is naturally increased by more 

 frequent supports, but need not be regarded as prohibitive. 



The diagrams (PL VII.) which accompany this paper have 

 been prepared from actual drawings, wherein the foregoing 

 construction was applied to a rod of variable cross-section for 

 which the mathematical solution w ? as known. It is easily 

 verified that if 23 and p are given by the expressions 



takini 



-s=m 



and 



P = 



>. 



(20) 



J 



— so that fSo and p are terminal values, — then equation (t>) 

 will be satisfied by assuming that 



Yx 



(■'- 



iron 

 I 



21 



■"> 



