Professor Sylvester on Elimination. 379 



facts of this extraordinary, and, as far as I know, unprecedented 

 case, but I do not offer any theory to account for the pheeno- 

 mena. It is hardly possible to suppose that there is any local 

 peculiarity about these boilers, or the place where they are 

 situated, to occasion the highly electrical condition of the steam 

 produced in them ; and yet it is as difficult to suppose the fact 

 of high-pressure-steam being electrical, a general one ; for if it 

 were so, it could hardly, up to this time, have escaped observa- 

 tion. The conditions, therefore, under which steam becomes 

 electrical require to be investigated, and it is not unlikely that 

 the investigation may lead to important results. 



I am, Gentlemen, 



Your obedient Servant, 



Bentham-Grove, Gateshead, H. L. PATTINSON. 



October 19, 1840. 



LVII. Note on Elimination. By J. J. SYLVESTER, F.R.S., 

 Professor of Natural Philosophy in University College, Lon- 

 don*. 



HP HE object of this brief note is to generalize Theorem C 2. 

 * in my paper on Elimination which appeared in the last 

 December Number of this Magazine. The Theorem so ge- 

 neralized presents a symmetry which before was wanting. 

 Here, as in so many other instances, the whole occupies in the 

 memory a less space than the part. 



To avoid the ill-looking and slippery negative symbols, I 

 warn my reader that I now use two rows of quantities written 

 one over the other, to denote the product of the terms re- 

 sulting from taking away each quantity in the under from 

 each in the upper row. 



Let h l h z ...... h n be the roots of one equation of co- 



existence. 

 /fj & 2 ...... k m of the other. 



And let the prime derivative of the degree r be required. 

 Take any two integer numbers p and <?, such that p -f q = r. 

 The derivative in question may be written, 



n \ 



, - _ - \ 2 .- ^p+ip+z"' n ^ 

 I ( x - ;/ i x h* - -n_V| V^i 3 V Ug+i k q +2...k m ) \ 

 VT-*I .x-k^.x-kj /*, h a ..h p \ /^ k v ..k 9 \1 

 \ WM ^p+2-A/ vi4-l*f4*"*/ 



N.B. Whatever p and q be taken, so long only as p + q 

 * Communicated by the Author. 



