60 Mr. Murphy on the Roots of Equations. 



of the same piece when the flame was fed with cold air ; but 

 I did not succeed in melting it. A piece of platinum foil, 

 however, showed signs of fusion upon its edges. By heating 

 the current of air to a still higher degree, by exposing it in 

 its course to a larger and a hotter surface, I should not despair 

 of accomplishing this object. 



Indeed, since I have had the good fortune to show that the 

 highest temperature of our furnaces does not, probably, ex- 

 ceed 3500° Fahrenheit, instead of 22,000°, at which it has 

 been, till very recently, estimated *, it is easy to understand 

 how a supply of air at 600° or 700° may increase their effi- 

 ciency ; and that a like augmentation of temperature bears a 

 considerable proportion to the melting point of cast iron reck- 

 oned at 2800°, which would be perfectly insignificant if the 

 same point were 18,000°. 



I remain, my dear Sir, yours very truly, 



King's College, Dec. 6th, 1832. J. F. Daniell. 



XI. On the Existence of a Real or Imaginary Root to any Equa- 

 tion. By R. Murphy, Esq. M.A. Fellow of Caius College, 

 Cambridge f. 



LET f(x) = be any given equation : put x = p + q s/ 1- 

 Then giving to p and q all possible values, there must be, 

 amongst the values of/* (#) which result, some one exactly = 0. 



For if not, if we reject all the imaginary results, there must 

 be some one amongst the real ones nearer to zero than any 

 other ; let the values of p and q be then P and Q, and let 

 R be the value of the function. 



Let h be an indefinitely small quantity, then changing P 

 into F + h, R would be changed into R + A h n (retaining only 

 the lowest power of h). 



But if we change simultaneously P into P + h cos — , and 

 Q into Q + ^, sin — , the whole increment to n or P + Q */ — \ 

 is then ^[cos \- V — 1 sin — ) ; and consequently f(x) 



(_ _ n 



cos — + s/ — 1 sin r-r ) 

 n n i 



= R-AA". 



* See Prof. Daniell's papers on his new Register Pyrometer, in Phil. Mag. 

 and Annals, vol. xi. ; and Phil. Mag. and Journ. of Science, vol. i. — Edit. 

 f Communicated by the Author. 



