of Beams and the Whirling of Shafts. 85 



similar to continuants. When expanded they contain only 

 even powers of the i/r's, so that the negative signs of the 

 latter may be changed to positive. They are obtained on 

 the hypothesis that sin K/ r is not zero for any value of r. 

 The solutions when that hypothesis is not fulfilled can, how- 

 ever, be obtained by considering the limiting ratios of the 

 functions involved when they become infinite, and these 

 solutions are important in the case where all the bays are 

 •equal. 



Case I. — Ends simply supported. 

 The period equation is 



A n = A [h, 1-2 • • Q = 01 + 02, f 2, 



J ^2 ? 02 + 03, ^3 



0, ^ 3 , 03 + 04 



= 0, 



0»-3-f-0n-2> ^i-2, 

 -v|r n _ 2 , n _2+0»-l, >K-1 

 0, ^„-l, ;i _l+0„ 



which can be evaluated by the formula 



A ? ~(0,_ 1 + n )A n _ 1 + ^ 2 n _ 1 A,_ 2 =O. 

 Particular cases : two bays, <^ 1 + <^ 2 = ; 

 three bays, (0 X + (/> 2 ) (0 2 + 3 ) — ^i = 5 



four bays, 



(01 + 2 ) (02 f" 03) (03 + 0l) - t/ (03 + 0J ~ ^(01 + 02^ = 0. 



Case IT. — One or both ends fixed in direction. 



The period equation may be derived from that in Case I. 

 by regarding an end fixed in direction as the limit of an 

 extra bay when the length of that bay is indefinitely 

 diminished. 



When both ends are so fixed the equation is 



01, ^i, 

 yfr , 1 + 2 , ^, 



0, *^ 2 , 02 + 0; 



= 0. 



jE> M _ 2 + 0»-l, ir n -h 

 ^r»_l, 0„_i+0 n , -f. 



0, 



^n, 0» 



