TORSIONAL OSCILLATIONS OF WIRES. 445 



To these expressions we have to add the outward pull of groups which break only 

 between r<p and rO. This is 



[ 2vrdr.\kv(rd-r<l>Y (13) 



J L 



By integration of the expressions (10), (11), (12), and (13), and by supposing, as for- 

 merly, that the forces act at a distance a from the axis, we find that the total inward 

 force is 



Llcva* { 2m [> * T- ($ _ *)*2_L- - <p 2 2 , * ,. - (0 - </>) 2 \ + 2nka*<t>\^ - \vaf\ 



5 ( i \_m m + lj nn + l ^+im 1 (m + \) ) Li 5 J 



if we take account of the pull (6) due to unbroken groups. This can be put in the 

 form 



hrkKa*(a<l>) -hrkvatia^) 2 ± ^-LrJcva^a^S \ . . '. (14) 



which reduces, when we put m = 0, to the expression (7) applying to the second half of 

 the inward motion. 



The points ( 1— — jrd are points such that, in the intermediate ranges, the multi- 

 pliers of the second and third terms in (14) remain constant. The sudden changes in 

 the magnitudes of these terms are equal and opposite. For, when (p reaches the value 

 /aO/^fx + l), X having become zero in the expression 1 +/j. + \, fj. is to be suddenly 

 changed to v — 1 in the affixes of the summations, so that the second term is suddenly 

 increased by the amount 



hrkua*(a? I*jp\± 9 = Lw^tf 2 )— 1 — 

 5 V 0" + l) 2 Jfi 2 5 V+l) 2 



which is also the decrease of the third term. Thus the force varies continuously. 

 The amount by which (14) differs from (3) at any definite value of the angle is 



- tphvaA % — 9 . a0 2 - S— , . a 2 <£ 2 . 

 o L i in im J 



This is therefore the continuously varying expression for the defect of the inward force 

 at a given stage in the inward motion from the inward force at the same stage in the 

 outward motion. 



The limiting boundary of the space included by the series of ellipses represented by 

 equating (14) to zero indicates the general relation between torsion and set when v is 

 constant. These ellipses intersect consecutively at points where 2<fi = 6, 3(p = 20, 

 4<£ = 30, etc. At these points the rate of variation of set with torsion changes suddenly. 



