OF FLUIDS ON THE MOTION OF PENDULUMS. [47] 



required in the calculation of the functions M , &c, and of the products LM , &c, in order to 

 ensure a given degree of accuracy in the result. The calculation by the descending series 

 (113) is on the contrary very easy. 



It will be found that the first differences of TO 2 // and of m 8 (k - 1) are nearly constant, 

 except near the very beginning of the table. Hence in the earlier part of the table the value of k 

 or k' for a value of TO not found in the table will be best got by finding VH-k - TO" or OT 2 k' by 

 interpolation, and thence passing to the value of k or k'. Very near the beginning of the table, 

 interpolation would not succeed, but in such a case recourse may be had to the formula? (103), 

 (104), (105), the calculation of which is comparatively easy whem TO is small. It did not seem 

 worth while to extend the table beyond OT = 4, because where TO is greater than 4, the series 

 (113) are so rapidly convergent that k and k' may be calculated to a sufficient degree of accu- 

 racy with extreme facility. 



38. Let us now examine the progress of the functions k and k' . 



When TO is very small, we may neglect the powers of TO in the numerator and denominator 

 of the fraction in the right-hand member of equation (105), retaining only the logarithms and 

 the constant terms. We thus get 



k + v/ - 1 k = 1 » 



4 



whence 



w 2 (fc-i) = — L**a m * k ' = ttv ■ • • ( ]15 ) 



Mi) "♦($)' 



L being given by (102) and (104), or (104) and (106). When TO vanishes, L, which involves the 

 logarithm of TO -1 , becomes infinite, but ultimately increases more slowly than if it varied as TO 

 affected with any negative index however small. Hence it appears from (115), that k — 1 and 

 k' are expressed by TO -2 multiplied by two functions of TO which, though they ultimately vanish 

 with TO, decrease very slowly, so that a considerable change in TO makes but a small change in 

 these functions. Now when the radius a of the cylinder varies, everything else remaining the 

 same, TO varies as a, and in general the parts of the force F on which depend the alteration 

 in the time of vibration, and the diminution in the arc of oscillation, vary as a 2 k, a 2 k', respec- 

 tively. Hence in the case of a cylinder of small radius, such as the wire used to support a 

 sphere in a pendulum experiment, a considerable change in the radius of the cylinder produces 

 a comparatively small change in the part of the alteration in the time and arc of vibration 

 which is due to the resistance experienced by the wire. The simple formulae (115) are accurate 

 enough for the fine wires usually employed in such experiments if the theory itself be appli- 

 cable ; but reasons will presently be given for regarding the application of the theory to such 

 fine wires as extremely questionable. 



From TO = *3 or *4 to the end of the table, the first differences of each of the func- 



