240 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Adding (1) and (4), we have 
- hf'X m <l>o) d ( m <t>o) ~ Q<1> 0 f"(mcf> 0 ) d (m(t> 0 ) + m</>Q/\m<l> Q )d(m<f> 0 ) . (5) 
m=q to m—1 m—0 to m=l ra=0tom=l 
Adding (2) and (3), we have 
<f>of'( m <l>o) d ( m <f>o) - q<f>of'( mc f>o) d ( mc l>o) ~ ™<f>of'X m <l>o) d ( m <t>o) • ( 6 ) 
ra—\—q to m=l „ m = 0 to m = 1 m = 0 to ?n=l 
Adding (5 ) and (6), we get finally, on integration, 
- 2 qUfito) -/'(0)] + * 0 [/(2*o) “/(I - 4M 
= -2r$[f'(r0 0 )-f'(O)] + re 0 [f'(rd)-f.r(6 o -6)] . . . ( 7 ) 
and the same result holds if q be less than 1 — q n . 
§ 11. If the expression (7) were applicable, the second bracket becomes 
zero when 6 = 6 0 /2, and the motion would be simple harmonic with an 
amplitude equal to the first maximum angle attained below the value 0J 2. 
Such a condition does not obtain in nature ; but it is not impossible that 
the kind of action under discussion may be fundamentally concerned in 
the process of “ accommodation ” described by G. Wiedemann {Phil. Mag., 
1880), according to which a wire, if twisted to and fro a few times between 
a definite positive and negative angular limit, obeys Hooke’s Law much 
more closely than before within that limit. 
But the form of the second term in the second bracket precludes the 
possibility of / being a power of the angular distortion, as it must be 
if the energy loss is to be practically representable as proportional to a 
power of the angular distortion. That term arises from the lower limit 
in the first term of (6). The simplest way of retaining the general mode 
of treatment followed, while getting rid of that term, is to modify the 
postulates in such a way as to add a term ^>o/ // ( m 0o) c ^( m ^ ) oX wi = 0 to 
m = 1 — q. This implies the assumption that the total back pull is less 
than that given above. Such a diminution might be due, for example, to 
the members of the groups contributing to the expression (3) acting 
differently when the reversed shear exceeds some value (m+gq)<p 0 , say 
where g< 1 ; or it might be due to the interaction with other groups. 
Instead of the expression (7) we should then get 
- 2 r6[f(r$ 0 ) -/( 0)] + rO o [f(r0) -/( 0)] . • . (8) 
§ 12. If we suppose that the commingling of effects alluded to at the end 
of § 7 is felt to the very core of the wire, the whole moment of the force 
due to unit length of the cylindrical shell then becomes the integral, from 
r = 0 to r — a, of the quantity 
- 8irk6r*dr\:f(r0 0 )-f'(Qy] + 4 t vh^dr0 0 [f(r0) -f(0)], 
