502 EEPOBT— 1889. 



Therefore 



h(h 



l + lr + k{k . 



ft 

 is one of the values of (1 + .r)'-. 



(ii.) We now assume the theorem proved for any positive exponent. Let n be 

 positive throughout, and, as before, let .r be numerically less than unity. 

 Since 



(l+.^•)"'.(l + .^■)" = (l+•^•)'"■*■"; 



therefore the product of 



1 + mx + - ^ — ^.r- + . . . . and 1 + nx + ^ ., — ■^" + • • • . 



is always 



-, , X (m + »i)(wi + M — 1) o . 



\ + {in + n)x + - il-g ^•^"■+ • . • 



when m is positive ; therefore, also, when ?« is negative. 

 Let ?»= — n ; then 



(l + (-«>+ -«(-»-lV + . . . .) (l + .r)» = l; 

 therefore 



7. On the JExteiisibility of Liquid Films, 

 By Lord Ratleigh, LL.I)., Sec.B.S. 



8, On Hysteresis in the relation of Strain to Stress. 

 By Professor J. A. Ewing, F.E.S. 



It is now well known that when an iron wire is subjected to the alternate 

 application and removal of tensile stress, many times repeated, certain of its qua- 

 lities which are affected by the changes of stress exhibit hysteresis with regard to 

 the changes of stress. If the load is cyclically varied between definite limits these 

 qualities do not have the same values at corresponding intermediate points during 

 the application and removal of load ; there is a hysteresis or lagging in the change 

 of quality which in some cases appears to be of a static character — that is to say, to be 

 independent of the time-rate of variation of stress. Conspicuous instances of this 

 action are seen in the changes of magnetic and thermo-electric qualities under 

 changes of stress, some of which have been described by the author in former 

 papers.^ 



It is natural to look for an effect of the same kind in the extension and retrac- 

 tion which the wire undergoes. We should expect that, after the change of loads 

 has been frequently repeated so that a cyclic regime is established, the wire will, 

 for any value of load intermediate between the two extremes, be longer during 

 unloading than during loading. Evidently, if such an effect exist, it must be 

 small, as it is well known that the proportionality of strain to stress which is ex- 

 pressed by Hooke's law is at least approximately exact. 



Sir W. Thomson's experiments on the damping of torsional vibrators have long 

 ago shown that an action of the kind spoken of" occurs in quickly performed cycles 

 of torsional strain. But it does not seem to have been looked for in slow cycles 

 of longitudinal pull. 



» Phil. Trans. Roy. Soc. 1885, 1886. 



