88 Mr. E. R. Darnley on the Transverse Vibrations 



cases^differ considerably from those of Dunkerley, as will be 

 seen'from the following table : — 



Table I. 



Value of l . 



^2 



Least value of IU, in radians. 



Dunkerley' s results, 



Author's results. 



1 



4 



i 



3-6056 



3-5101 



3-32S2 



3695 ) 

 3-6923 j 



3 643 



3-565 



The results in the last column, except the result 3*6923, 

 have been obtained by double interpolation from the annexed 

 tables. The result 3*6923 has been obtained in a more 

 accurate way by the use of the large tables of hyperbolic 

 functions issued by the Smithsonian Institution, and indicates 

 the amount of error to be expected in the use of the short 

 tables. It appears that the method of approximation adopted 



by Dunkerley is accurate only when y is very small. 



Dunkerley's experimental results in these cases differed 

 from his theoretical results by percentages — 2'4, +0*5, and 

 + 6*2, or an average of 3*0 per cent. The new results would 

 apparently give differences of -{-2*4, +7*1, and +7*6, or an 

 average of 5*7 per cent. 



When the bays tend to equality, it will be seen from the 

 graph that the least solution tends to 180°. 



§ 9. The equation for three bays is 



(01 + fa) W>2 + 03)=f2 2 , 



and it will be easy to find the least value of K in any 

 particular case. 



Take the case of l x : l 2 : Z 3 = l : 2 : 3. 



If any solution exists for which K/ 2 is less than ir, cj) 1 will 

 be positive, and $ 2 will be numerically greater than yjr 2 

 Hence, to satisfy the above equation., c/> 3 must be negative 



