AND ON NEWTON'S METHOD OF CO-ORDINATED EXPONENTS. 627 



To avoid the pointed branch, we are at perfect liberty to invent a symbol : we may say, 

 let e be understood as e (cos + sin . -y/- l), a perfectly distinct thing from e x or 

 e (cos 27r + sin 27r . •%/" l), &c. Such a distinction I not only admit, but upon occasion 

 contend for : but granting y = e * to be fully represented by one continuous branch ; y=e{° by 

 a system of isolated points having ,v = ^p; y = e/ by another system having x = ^p, &c. ; p 

 being any integer, positive or negative— the curve y = e x (cos 2irmx + sin 27rmx . ^/ — l), or 

 y = c x in the ordinary sense, collects all the cases, infinite in number, and adds the pointed 

 branch to the continuous curve. 



The discovery of equivocal value in the inverse functions of common algebra was accom- 

 panied by what is but the expression of the result in another way, the discovery of equivocal 

 form in the direct functions. Thus the double square root of x implies the double form of 

 a? 2 , (+ a?) s and (— a;) 2 . The triple distinction of +,0, -, is connected with the similar dis- 

 tinction of expressions into logarithmic, common algebraical, and trigonometrical, in whichever 

 order we read them. Each has its direct and inverse forms, as in e x and log a;, x m and x™> 

 tan x and tan -1 *. If univocal character be the proper attribute of a direct function, as 

 opposed to an inverse one, then x m (m integer), and tana;, are direct. But both e* and 

 logo? are equivocal, and e* has an equivocal character resembling that of xm, while log a?, in 

 this respect, resembles tan -1 a\ Moreover, the nature of the equivocal values of logo; and 

 e x depends upon the value of x: in other respects, log x is analogous with the simple power 

 of x, x m , and e* with Xm ; while log a? is analogous with tan _1 a', and e* with tana?. Thus, 

 x being the positive value { e i2p+ly ' 2q , log (— a?) = log a;, in its real value, while x and — x 

 are the two values of the exponential. 



In order fully to trace the curve F (x, e**, e**, log ^a?,...?/) = 0, we must first, without 

 attending to conditions of punctuation, allow all the multiplicity of branches which would be 

 given by F (%, i^/P, ±^/Q, R 2 , ... y) = 0, and then, selecting the branches in which one or 

 both of — \/P, — \/Q., are used, punctuate them by only allowing y to exist when one or 

 both of (px and \px have the forms (2p + 1) : 2q. If, by the general process, we desire 

 to ascertain the singular points, then, rationalizing F (x, -\/P, \/Q, ...) = into/ (a?, y, P, 

 Q, R\...y) = 0, we must fully ascertain the singular points of f(x, e 2 **, «**?, log -^x, ... y) = 0, 

 and then ascertain the pointed branches by reference to the original form, having previously taken 

 only positive values of e 2 *'* and e^ 1 , and considered log (—a) as log a. When ^a? is negative, 

 the branch thence arising is punctuated by the condition -^x = 6 t2p+1) : 2g . The so-called abrupt 

 termination is always the point of union of a continuous and pointed branch ; and the salient 

 point where two tangents can be drawn to one point, is but a double point at which two 

 mixed branches meet. 



The rejection of the pointed branch is an abandonment of geometrical algebra, and a 

 return to algebraic geometry : but the stand is made too late, and, in fact, the place at which 

 it shall be made is a matter of taste, unless it be made upon a definition. A system which 

 admits an isolated point, any given number of them, any smallness of distance between isolated 

 points, certainly draws upon algebra for its definitions, and must end with the pointed branch. 

 An application of the Horatian pitchfork to any symbolic result will only be another fulfilment 

 of the Horatian prophecy. A. DE MORGAN. 



University College, London, April 16, 1855. 



