164 Mr. A. Stephenson on the Frequency Ranges of 



Multiply by r, differentiate with regard to r, and divide by 

 r, and (25) becomes 



B« 2 



l*p_ =g pa 9 !3W rf , +v 3^lW, (26) 



^ B^ Jo ^^ r ^ ? ' r or dtQZ r or 



Substitute 



-^k- 1 =Mo(wr)Z, 

 r or 



where Z is a function of z alone. Then 



i a r a i ao) = y ibw ( 1 a law 

 ^a^ a^ r a^ a^r a^ ra^^ a^ 



__ z / a 2 J (mr) [ i aJ (mr) \ 

 \ dr 2 r ~dr f 



= — m 2 J (mr)Z. 



Hence J (mr) cancels out of equation (26). If we assume 

 a time-factor e X(Tt we have for Z 



a^Z-gm? f Z^ + Wo-|^, =0, . . . (27) 



Jo o^ g 



which is the same as equation (7). 



VIII. On the Frequency Ranges of Non-generating Force 

 exerting Cumulative Influence. By Andrew Stephenson*. 



1. T)ERI0DIC non-generating force acting on a system 

 JL in oscillation about a position of stable equilibrium, 

 exerts a cumulative action in intensifying or diminishing the 

 amplitude if its frequency is contained within any one of a 

 number of ranges lying in the vicinity of 2/u,, 2/j,/2,[2/jl/3 ..., 

 where /j, is the natural frequency of the system f. i 



The limits of the leading ranges have been obtained for 

 small non-generating force. It is our object here to give a 

 general method of finding the ranges in magnitude and 

 position when the force is finite, and in particular to obtain 

 the numerical values of the range limits about the double 

 frequency for various intensities. 



* Communicated by the Author. 



t "On a Class of Forced Oscillations," Quart. Journ. of Mathematics, 

 No. 168, 1906. 



