800 EXPLORATION GEOPHYSICS 



of motion. The Lagrangian equations for various types of electric seismometers will 

 be found in the article by Scherbatskoy and Neufeld.f 



The equations of motion of the electromagnetic reluctance seismometer may be 

 obtained by substituting the expressions for T, V , and D as given by Equations 157, 

 158, and 159 into the Lagrangian equations (160) and (161). In carrying out this substi- 

 tution, certain terms will be zero and certain other terms will be constants which for 

 the sake of simplicity may be set equal to zero. For example, Tt (equal to Tei + 

 7^,2 + Tea) is a function of j only. Hence, the derivative of T« with respect to s van- 

 ishes; i.e., 



Also -XT- I —r^ I is a constant which may be set equal to zero, and -r— (Li),=o is also 

 OS \ as / OS 



«=0 



a constant which may be set equal to zero. However, it will not be assumed that 

 I -T— j is zero, because that would be equivalent to assuming that there is no 



viscous drag. As a matter of convenience in notation, set 



(f)„-^» (163) 



Various derivatives of the energies with respect to the electric charge q and the 

 current i (equal to q) likewise vanish. Thus Tm equal to J^«z/ is not a function of 

 1 or q and T, = Tei + Te2 + Tes is not a function of q. Hence, 



^^^ = ^ = ^ = (164) 



ot oq oq 



m\ _ dT\ ^ dTe, _^ dTe^ ^ dTe, 

 9 7 3i 9 J 9 J 9j 



?J^ = 0+(Li).=ot + iC„5 (165) 



oi 



(LO.=o = L (166) 



^=Li + K„,s (167) 



ot 



01 oq 



On substituting into the Lagrangian equation (160), the expressions for T (Equa- 

 tion 157) for V (Equation 158) and for D (Equation 159), and making use of Equa- 

 tions 162 and 163, the following relation is obtained : 



d dTm , d dTe dTm 97, , dVm ,9 .^ m r> \ — ^ 



-dT^r-^-dT-^—^—^'^-^^'^^^-''^'^-^ 



Also 



Setting 



Also 



m's - K^i + -^ + r,ns = / ( 169 ) 



Cm 



d's 

 where s = -j^ 



Similarly on substituting the expressions for T, V and D and making use of 

 Equations 164 through 168, the Lagrangian equation (161) becomes: 



_d_dTjn j^A-'^Ll — '^—^Lh ,9^, 9 (An + De) _ ^ 



dt 9t dt 9i dq dq dq dq 



t S. A. Scherbatskoy and J. Neufeld, "Fundamental Relations in Seismometry," Geophysics, 

 Vol. II, No. 3, July, 1937, pp. 192-212. 



