TORSIONAL OSCILLATIONS OF WIRES. 619 



instead of 



y n (x + a) = b (2) 



Now we have 



-dy — -jy n+l dx. 



Hence, if, dividing y' by Jc, we reduce the equation, supposed to have been determined 

 in the form (l), to the form (2), we see that the quantity k gives the value, in terms 

 of the units used in (1), of an angle at which the rate of diminution of the oscilla- 

 tions is totally independent of the value of n. So also, since the loss of energy per 

 oscillation is proportional to yoly (see former paper), this angle, which we may call the 

 Critical Angle, is such that the loss of energy per oscillation is independent of n. At 

 larger angles there is a greater loss when n is large than when it is small ; at smaller 

 angles there is less loss when n is large than when it is small. 



Now a is increased by increase in the magnitude of the initial oscillation and also by 

 fatigue. Hence we see that, at smaller angles than the critical angle, fatigue causes 

 a decrease in the loss of energy per oscillation, and, at larger angles, fatigue causes an 

 increase in the loss. Fatigue has no effect on the rate of loss of energy per oscillation 

 at the critical angle. 



Possible Explanation of the Existence of a Critical Angle. 



Greater stability of a given group of molecules in a molecular system may consist 

 in increase of (1) attractive forces amongst its components at a given distortion of the 

 whole system, or (2) range of distortion of the whole system before rupture of the given 

 group occurs. Either of these results may ensue upon a rearrangement of the con- 

 stituents of the given group. 



Decrease of (1) or (2) may occur provided that it is more than balanced by increase 

 of (2) or (1) respectively. Thus there are five possible cases. 



If increase of (l) alone occurred, there would, in all cases, be greater loss of energy 

 in an oscillation of given magnitude. 



If increase of (2) alone occurred, there would be less loss of energy in an oscillation 

 of given magnitude. 



If increase of (1) overbalances decrease of (2), greater loss occurs in all cases; if 

 increase of (2) overbalances decrease of (1), less loss occurs in all cases. 



Under none of these conditions can a critical angle occur. 



When increase of (l) and (2) occur together, a critical angle exists when the effects 

 balance as to loss of energy. At smaller angles increase of (2) must preponderate ; at 

 larger angles increase of (1) must preponderate. 



A critical angle would also exist if, below it, increase of (2) overbalanced decrease 

 of (1), while, above it, increase of (1) overbalanced decrease of (2). This necessitates 

 that, at the critical angle, neither (1) nor (2) are changed. This could only mean that, 

 at that angle, all configurations were equally stable. But, apart from that, it seems 



VOL. XXXVIII. PART III. (NO. 18). 4 R 



