X 

 is also equal to 



REPORT ON CERTAIN BRANCHES OF ANALYSIS. 235 



consideration of all the circumstances under which they pre- 

 sent themselves in symbolical results. In order, therefore, to 

 determine some of the principles upon which those interpreta- 

 tions must be made, it will be proper to examine some of the 

 more remarkable of their symbolical properties. 



and from whence it will follow that the non-introduction of such can pro- 

 duce nothing wrong." Thus, x^ + ax + b, which is equal to 



{^/(. + i)'W(^-0} 



whatever be the value of the quantity /3 ; a conclusion which enables us to 

 reason upon real quantities and to make /3 = 0, when the primitive factors 



are required. Similarly, if instead of 2 = V' '^^ suppose 



^ ^ = V — R , and if uistead of = «, 



2 2 a/— 1 



we suppose ^ ^ ^ = x — R, we shall find, whatever /3 may be, 



'^ 2VjS— 1 



^* v'p-i ^_ — ^t ^_ ^ 1^ — I ^x — R), a result which degenerates into 



the well known theorem e'^-^ =y + V—1 a-, if /3 = 0. Many other ex- 

 amples are given of this mode of porismatizing expressions, (a term derived by 

 Mr.Gompertz from the definition of porisms in geometry,) by which operations 

 are performed upon real quantities which would be otherwise imaginary : 

 and if it was required to satisfy a scrupulous mind respecting the correctness 

 of the real conclusions which are derived by the use of imaginary expressions, 

 there are few methods which appear to me better calculated for this purpose 

 than the adoption of this most refined and beautiful expedient. 



The second tract of Mr. Gompertz appears to have been suggested by 

 M. Buee's paper in the Philosophical Transactions, to which reference has been 

 made in the text : it is devoted to the algebraical representation of lines both 

 in position and in magnitude, as a part of a theory of what he terms func- 

 tional projections, and embraces the most important of the conclusions obtained 

 bv Argand and Franfais, with whose researches, however, he does not appear 

 to have been acquainted. I should by no means consider the process of rea- 

 soning which he has followed for obtaining these results to be such as would 

 naturally or necessarily follow from the fundamental assumptions of algebra : 

 but it would be unjust to Mr. Gompertz not to express my admiration of the 

 skill and ingenuity which he has shown in the treatment of a very novel 

 subject and in the application of his principles to the solution of many curious 

 and difficult geometrical problems. 



