On the Effect of Internal Friction on Resonance, 177 



course, occurs iu the lunar and planetary theories, but with this 

 difference : there the difficulty is introduced by the method of 

 solving the differential equation, and is avoided by modifying 

 the first approximation to a solution ; here it is inherent in the 

 differential equation, and can only be avoided by making that 

 equation express more completely the physical circumstances of 

 the motion. One or more of the assumptions on which the dif- 

 ferential equation rests is invalid. We must look either to 

 terms of higher orders of smallness, to resistance of the air, or 

 to internal friction. With the modifications due to the last 

 cause we are now concerned. 



The approximate effect of internal friction is probably to 



add to the stress E -^-, produced by the strain ~ when the parts 



of the body are relatively at rest, a term proportional to the rate 



at which the strain is changing ; so that the stress when there 



fd£ (fit \ 

 is relative motion will be E ij- + k -~~ Land our equation 



of motion becomes 



The solution of this equation will contain two classes of terms. 

 First, a series corresponding to those under the sign of summa- 

 tion in (2), which principally differ from (2) in the coefficients 

 decreasing in geometrical progression with the time, the highest 

 fastest, and in the total absence of the notes above a certain 

 order as periodic terms; these terms we may consider as wholly 

 resulting from the initial conditions, and as having no perma- 

 nent effect on the motion. Second, a term corresponding to the 

 first term of (2), and which expresses the state of steady vibra- 

 tion when work enough is continually done by the forced vibra- 

 tion of the extremity to maintain a constant amplitude. The 

 investigation of this term is a little more troublesome, because the 

 motion is periodic, the effect of friction being to alter the motion 

 in a manner dependent on the position of the point, not on the 

 time, and equation (3) cannot be satisfied by a sine or a cosine 

 alone of the time. 



Assume f = <£(#) sin m£ + 'vjr (#) cosm/, 



or a series of such terms, if possible, each pair satisfying equa- 

 tion (3). Substitute in the equations of motion, and equate 

 coefficients of sin mt and cos mt, 



a a (0" - kmf") = - m 2 (£, \ 

 « 2 (f+k(()") = -m^J ( ' 



Phil. Mag. S. 4. Vol. 45. No. 299. March 1873. N 



