2-10 Prof. J. D. Everett on Dynamical 



k* + k— - 1 =0 (32) 



as s 



Neglecting the first term (which is small compared with 

 the others), we have 



as the approximate value of one root. It is the value obtained 



by putting 5 = in the general quadratic, and is identical 



with the value of k obtained by supposing the point of support 



constrained to vibrate like a pendulum of length a. The 



other approximate root is 



, b — a 

 k = : 



as 



for the product of the two roots must be — l/s. 



It will be noted that the first root makes the excursions of 

 the two pendulums comparable with one another ; whereas 

 the second root makes the excursions of the upper very small 

 compared with the lower. 



The closest approximations to the values of H 2 are got by 

 employing in connexion with the first root the formula 



and in connexion with the second root the formula 



The specifications of the two modes will accordingly be 

 First Mode. Second Mode. 



a — o sa I 



> ■ ■ ■ (33) 



a\ b — a/ b\ a — b) J 



The first mode nearly agrees in period with a pendulum of 

 length a, and the second with a pendulum of length b. 



The two values of k are obviously opposite in sign. The 

 positive k always makes D. 2 less than the lesser of gja and g/b, 

 and the negative k makes it greater than the greater, in 

 accordance with the general rule. When a is infinite one 

 value of H 2 is (l + s)g/b, and when b is infinite one value 

 is (l + s)g/a, as previously found for the general case. 



§ 12. The results obtained in the preceding section agree 



