1912-13.] Torsional Oscillations of Metallic Wires. 241 
where a is the external radius of the wire, and 2 k is the Hooke’s Law 
coefficient. Using the notation /= H'", this becomes 
- 8 " { - 4s> h> ‘ u + 1 
+ 4t rka^iaey^'iaO) - 3(aO)- 2 E'\aO) + 6(a0r 3 E\a0 ) - 6(atf o )- 4 [H(a0) - H(0)] } 
- 7r&a 4 H""(O)(0 o -20) (9) 
If we now postulate a solution 
H(.r) = Ax p , 
we have H(0) = 0, and also H""(0) = 0 since p = n + 5 > 4. Then (9) becomes 
- iirkAa\p- l)(p-2)(p- 3)p.2(ag- 4 -^ 0 (fl^- 4 ] . . (10) 
which is Hooke’s Law modified by a term varying as a power of the dis- 
tortion. It therefore leads to the observed law of dissipation of energy. 
And it also gives the observed experimental results regarding the law 
of oscillation (§ 5). In the inward motion from a maximum shear jj.(p 0 , the 
motion is simple harmonic to the position of set, beyond that to the initial 
zero, and beyond that to an extent which may reach the range (1 — /m)(p 0 /r 
or not, according to the nature of the action which changes (7) into (8). 
If there is any inferior limit to shear at which commingling of effects 
ceases, at least one type of groups which dominated the action ceasing to 
break, a sudden change in the power of the distortion could occur when the 
range of oscillation diminished to the corresponding value. Thus it may be 
that a sudden change in the value of n in the equation y\x + a) — b is due 
to the existence of a dominating type of groups which is effective above a 
certain range of oscillation and becomes ineffective below it. The tempera- 
ture or stress effects described by Mr Ritchie would then be explainable, 
by the production or destruction of one type of groups. The shifting, with 
change of temperature, of the limiting range of oscillation at which the 
value of n suddenly altered, seems to indicate a change of the limiting 
shear at which one type of groups breaks down. This is naturally to be 
expected in the neighbourhood of a temperature at which a change in 
crystalline form takes place. 
§ 13. The necessity for the view that commingling of effects takes place, 
even in the sense of the sharing of the action between groups in which 
the strain is sufficient itself to effect rupture and those in which it is not 
sufficient, at once appears, if we integrate (8) only throughout the region 
in which the former condition holds. The limits for rO are then <pj 2 and 
a0, while those for r6 0 are (p 0 6J26 and a0 0 . Instead of (10) we then obtain 
MA(p-l)(p-2)(p-3)l { 
0 L 
K 
m 
(aoy 
2 
]} 
16 
( 11 ) 
VOL. XXXIII. 
