'40 



Mr. J. R. Airey on the 

 Load less than mass of chain 



Mode of 

 vibration. 



Yalue 



Of u. 



Length of 

 chain. 



C11JS. 



Time of 

 vibration. 

 Calculated. 



Time of 

 vibration. 

 Observed. 



1 

 2 



4 

 5 



148-8 



2-297 

 •639 



2-274 

 •638 



1 



o 



1 

 2 



do. 



2255 

 •715 



2-246 

 •715 



1 



2 

 3 



1 

 3 



do. 



2-211 



•772 

 •428 



2-200 

 •766 

 •426 



1 

 2 



3 



1 



8 



do. 



2-128 

 •856 

 •506 



2-114 

 •850 

 •500 



1 



2 



3 



1 



15 



do. 



2-092 

 •883 

 •537 



2-076 

 •877 

 •531 



1 

 o 



3 



4 



1 

 24 



do. 



2072 



•890 

 •557 

 •386 



2-058 

 '882 

 •552 

 •382 



(D) The general expression for the roots of the equation 



Y,flu) g Y,(t)-2Y 1 ( 8 ) YjM 

 J^X*)-*J,(*)-SJi(*)-J,(*)'. 



where A is greater than unity, can be obtained by following 

 the method adopted by Prof. McMahon* in finding the roots 

 of Bessel and other related functions. 

 Subtract log 2 — y from each sidef. 



Then N (\~) _ N,fe) 



j (A^o -j 3 (*y ' ' ' 



where ?K n [z)~Y.{z)-(tog*-v)Mz). 



• (3) 



Substitute the semiconvergent series for J Q (\z), N (X^), &c. 

 in equation (3) and put 



Qo(7i2-) = Rsin^ F (Xz) = Ucos6 . . . (4) 



Q 2 (,2r) = Ssin?7, T 2 {z) = 8costj. . . (5) 



* McMahon, Annals of Mathematics, 1895. 

 t Jahnke u. Emde, Funktionentafdn, 1007. 



