[104] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



series which are at first rapidly convergent, and which enable us to calculate the numerical 

 values of the integrals with extreme facility. These expressions were first given by M. 

 Cauchy, in the case of Fresnel's integrals, to which the integrals just written are equivalent. 

 They may readily be obtained by integration by parts, though it is not thus that they were 

 demonstrated by M. Cauchy. If now the above expressions be substituted for the integrals 

 in (195) the terms containing $ destroy each other, and for general values of t the most im- 

 portant term after the first contains t~l. Since however t is supposed to correspond to the 

 end of an oscillation, so that nt is a multiple of ir, the coefficient of this term vanishes, and the 

 most important term that actually remains contains only £~4. Hence neglecting insensible 



quantities we get from (195) 



A + AA c / 7T 

 J °S^7^ = ^V 2 - 096) 



We get from (194) by performing the integrations 



jL + A A„ / t r* • 



log = c \r — / sin nt (cos nt + sin nt) dt 



A *>n •*/. 



C / TT , . . 



= — V/ — <2nt + 1 - cos 2nt — sm 2nt>, 

 4rc v 2» l 5 



which becomes since nt is a multiple of tt 



. A a + A A c / tt 



lo g — -j — -ttV ^r- 2nt (197) 



A in v 2» 



We get from (196) and (197) 



whence 



*nt log dtt*4 = log^t^ = log^ +AJ ? + log 4 , 

 B A B A * A s A' 



, A + A A^ A,. 



log ■■--. = (int - I)" 1 log -j (198) 



A -*•"• *' "'*A' 



and the same relation exists between the common logarithms of the arcs, which are propor- 

 tional to the Napierian logarithms. Now Log A - Log A is the quantity immediately deduced 

 from experiment, and Log (A + A^„) - Log A Q is the correction to be applied, in consequence 

 of the circumstance that the motion began from rest. Instead of applying the proportionate 

 correction + (2nt - I)" 1 to the difference of the logarithms, we may apply it to the deduced 

 value of ySfjL, which is proportional to the difference of the logarithms. In Coulomb's 

 experiments 10 oscillations were observed, and therefore 2nt = 20tt, and (2nt - 1) _1 = 0.01617, 

 and the uncorrected value of y/j/ being 0.0555, we get 0.0009 for the correction, giving 

 yV = 0.0564. 



