428 Prof. Sylvester on Rational Derivation 



5. Dead gray metal from Terre Noire, near Lyons 7'014 



6. Grayish iron from Tollend, S. Staffordshire 7"144 



7. Mottled iron from Staffordshire 7'329 



8. White metal forced out from cast-iron pigs at Alais 7'339 



9. Whitish gray cast iron from Alais 7'404 



10. White cast iron from the Marley Iron-works, South Wales.. 7'407 



11. White cast iron from Alais, (fine granulated) 7*442 



12. Mottled iron from Staffordshire 7*451 



13. White crystalline iron 7*472 



14. Bar iron heated in a reheating furnace 7*487 



J5. Wootz from Bombay ". 7-508 



16. No. 12, as plate metal 7*513 



17. Cement steel made from English iron (highest concentration) 7*519 



18. Cement steel from Swedish iron (double bullet) middling hard 7*563 



19. Cast iron, silvery white, extremely large-grained, from Alais 7*582 



20. Cement steel from Swedish iron {Koop L.), highest conversion 7*621 



21. White-charcoal iron from Fourchambault (France) 7*709 



22. Cold-short bar iron from Staffordshire 7*753 



23. Cement steel made from bar iron, prepared from Staffbrdshire 



iron according to my method of puddling (highest conver- 

 sion) 7*755 



24. Swedish iron called Koop L. in soft gray leaves 7*761 



25. Iron melted by me with 3 per cent, of sugar coal (it hardened 



not at all) 7764 



26. Swedish iron, called double bullet, fracture fine-granulated . . 7*810 



27. Bar iron, according to my puddling process for steel (fine-gra- 



nulated) 7*841 



28. The same, the granulations interrupted by fibres 7*845 



29. The same, the fibres more developed 7'840 



30. No. 23. in cast-steel ingots 7-892 



31. The foregoing tilted into small bars for razors 7*900 



Of the lowest specific gravity, it will be seen, was gray bril- 

 liant cast iron from Vienne, department d'lsere in France, 

 equal to 6*898 ; and of the greatestspecificgravity appears tilted 

 razor-steel, prepared, according to my method, from English 

 iron at Mr. Solly's works at Dudley, equal to 7"900. 

 [To be continued.] 



LXIII. Memoir on Rational Derivation from Equations of Co- 

 existerice, that is to say, a neiio and extended Theory of Eli- 

 mination. By J. J. Sylvester, F.R.S.y Professor of Natural 

 Philosophy in University CollegCi London^. (Part I.) 



A NY number of equations existing at the same time and 

 -^^ having the same quantities repeated, may be termed 

 equations of coexistence : in the present paper we consider 

 only the case of two algebraical equations : 



x'^ + a^, a^-'^ + aciX'^--^ +,.,,., -\-am — 



A"' + b^ A'"-l + ^'2'2^"~^+ -f Z»„ = 



* Communicated by the Author. 



