168 Mr. P. Nicholson on the 



While examining the products of the above gentlemen's 

 manufoctory, I met with n substance adhering tn the interior 

 of the tube' leading from the retort, which is interesting from 

 its being an alloy of 92'6 of zinc with 7*4> of iron ; contrary 

 to the opinion expressed in chemical works. Its specific gravity 

 is 7-172 at fi2°; it is extremely hard and brittle: the fracture 

 shows broad facets like zinc, but of a duller grey colour, with 

 surfaces more rough and granular. Its situation perhaps fin'- 

 nishes us v/ith a hint, of a way to form alloys of metals which 

 are not miscible in the open iurnace. 



Yours very truly, 



WlLMAM HeRAPATH. 



XXXV. On the Transformation of Functions. By Mr. 

 Peter Nicholson*. 



To the Editors of the Philosophical Magazine and Journal. 

 ^ t "'HE following demonstration of the method of transforming 

 -*• the function of an miknown quantity x into another in 

 .T — e, or for extracting the root of equations, though not dif- 

 ferent in principle from that which I published in your Maga- 

 zine, September 1822, will perhaps be tbund to be not only more 

 explicit, but much more comprehensive, and better adapted to 

 the understandings of the general mass of readers. 



I shall therefore feel obliged by your publishing my en- 

 deavours to improve a subject so very interesting in mathe- 

 matical science. And at the same time I shall observe, that 

 the method of transforming a function, or of extracting the 

 root of an equation, is only a branch of a more general theorem 

 belonging to a new species of analysis, which I hope will 

 shortly aj^pear through the medium of your valuable Journal. 



5 Claremont-place, Jiukl-street. P. NiCHOLSON. 



Proposition. Theorem. 



If X = i> +c 



A.r +B =Av +B, 



A.r'- + B.r+C . . . =Av''-\-B,v +C., 



A,i'^ + BxH Cr+D = At;^ + By + C,u-f-D 

 &c. 

 Then will 



B.=B +^A 



B,= B,+^A C -C +fB, 



B,= B,-|-^A C,= C, + eB, D3=D + eCJ 



&c. Sec. &c. I &c. 



* Communicated by the Author. 



De- 



