TRANSACTIONS OF SECTION G. 751 



increases, the resistance of tlie material increases continuously up to the point of 

 fracture. Nevertheless, it is shown that, as .v increases, the relations between the 

 stretching force and the resistance pass from a condition of stable, through indif- 

 ferent, to unstable equilibrium. The law of plastic extension, y = ^(.i), as defined 

 by the curve, fixes mathematically the occurrence of the ' plastic limit ; ' and it fixes 

 also the ' breaking weight per square inch of original area,' which can have only one 

 value, and is easily found by graphic construction. The breaking weight of the tie- 

 bar as thus defined depends upon the plastic extensibility, in much the same way 

 that the breaking weight of a strut depends upon the elastic modulus. 



8. Examining next the possibilities of deformation in a bar of uniform or 

 nearly uniform extensibility, it may be seen that, as the plastic limit is approached, 

 the slightest irregularity in section or in extensibility tends to precipitate tb(} foi'- 

 mation of a contracted region, and ensures that beyond that limit the further 

 extension of the bar and the further contraction of area will be confined to the 

 same region. For stresses below the plastic limit the probabilities of deformation 

 mav be examined by considering the relative time-rates of extension at two elements 

 which may have been unequally stretched, and at first the tendency is theoretically 

 in favour of preserving the cylindrical form of the bar. But beyond the plastic limit 

 these conditions are reversed, and the tendencies are all in favour of precipitating 

 the most rapid contraction of area at the point where any contraction already 

 exists, and thus to pinch in still further the region where any contraction of form 

 has begun to show itself. 



The observed phenomena agree very well with these deductions, and may be 

 rationally attributed to the operation of the mechanical conditions named. 



9. Going back to the yield-point, the sudden elongation which here takes place 

 appears to be something difl'erent from plastic extension. Examined analytically, 

 the process is found to take place at somewhat different stresses in the different 

 segments, while in any one short element or segment it seems to be instantaneous. 

 If the yield is arrested midway and the bar examined, it is sometimes found that 

 the elongation has been quite completed in some segments, and not even commenced 

 in others. This irregularity is greater than anything met with in connection with 

 plastic extension, and may account for the variations of form observed in auto- 

 graphic diagrams at this point. When rhe process has taken place throughout, it 

 constitutes an elongation which is almost uniform tbroughout the bar, and after 

 the yield the segments fall into line and exhibit a nearly identical curve. As 

 regards the slight curvature below the yield-point, it may be doubtful whether this 

 indicates the commencement of yield in some short elements, or the commence- 

 ment of a plastic extension in the course of which the vdeld occurs as a separate 

 incident, 



10. The author recognises the insufficient nature of his experimental data, and 

 the need for further study and more refined measurements of the true curve of 

 plastic extension ; but judging from the results obtained, some formulfe ai-e suggested 

 as representing the probable law of plastic extension for such a ductile material. 



11. In the paper some further experiments are described which were made with 

 the object of measuring the extensions on a larger scale, and of testing some of 

 the propositions before referred to. 



3. On Tunnel Construction by means of Shield and Compressed Air, ivith 

 special Reference to the Tunnel under the Thames at Blackivall. By 

 Maurice Fitzmaurice. 



Since 1892, when Mr. George F. Deacon read a paper before the British Asso- 

 ciation on the construction by means of shield and compressed air of the Vyrnwy 

 Aqueduct Tunnel under the Mersey, other and larger tunnels have been constructed 

 and are now in course of construction in this country by the same methods. 



Three tunnels hare been thus completed under the Clyde just below Glasgow, 

 and tunnels at several points in Glasgow are being made in connection with the 

 Glasgow District Subway. A tunnel is also being constructed in Edinburgh by 



