38 REPORT—1846. 
certain irrational functions* of a as rational functions of 2, a, b,...¢. For 
instance, if the equation were a® + ax + se — 1)=0, it follows that 
V7{—x*=a+2. So that any irrational function of the form F (2 “1—z*) 
can be expressed rationally (F being rational) in # and a. 
From the given equation we deduce by differentiation the following, 
dxe=ada+fdb+...+yde, 
where a, 6, ... y are rational in a, a, b,...,¢. 
Let y be one of the functions which can be expressed rationally in a, &c., 
it follows that ydx=Ada+Bdb+...+Cde, 
where A, B, ... C are also rational in a, &c. 
The equation fa = 0 will have a number of roots, which we shall call 
jy +++, It follows that 
Yrdt, +e tyday, = 
{A,+..+A,}da+{B,+..+B,}db+...+{Cit..+C,} de, 
where the indices affixed to y, A, &c. correspond to those affixed to 2, so 
that y,, for instance, is the same function of 2, that y, is of 2. 
Now A, +... + A, is rational and symmetrical with respect to #,...#y, 
therefore it can be expressed rationally in the coefficients of f («) = 0, and 
therefore in a, b..¢c. We will call this sum R,, and thus with a similar 
notation for b, &c. we get 
Y, a2, + oe. + y, dk, = R,da+R,db+...+R.de. 
The second side of this equation is from the nature of the case a complete 
differential, and it is rational in a, }, c, &c.; it can therefore be integrated 
2. 
by known methods; and if we denote Y dz by )(a,), we get 
v(a,)+---+ ¥(¢,) =M, 
M being a logarithmic and algebraic function of a, b, &c., which we may 
suppose to include the constant of integration. 
) (x) is in general a transcendental function, while a, 6, &e. are necessarily 
algebraical functions of x,,..-, x, and the result at which we have arrived 
is therefore an exceedingly general formula for the comparison of transcen- 
dental functions. 
The simplicity and generality of these considerations entitle them to espe- 
cial attention: it cannot be doubted that the application thus made of the 
properties of algebraical equations to the comparison of transeendents will 
always be a remarkable point in the history of pure analysis, : 
A very simple example may perhaps illustrate what has been said, Let us 
recur to the equation 1 
w+ax + 5(a?—1)=0, . . * . ° e (1,) 
and suppose that an wig 
y V1 —a® 
Differentiating the first of these equations, we find that 
(Qa +a)dx+(«#+a)da=0. 
* It must be remembered that an algebraical function is either explicit or implicit: ex- 
plicit, when it can be expressed by a combination of ordinary algebraical symbols ; implicit, 
when we can only define it by saying that it is a root of an algebraical equation whose co- 
efficients are integral functions of 2. Thus y is an implicit function of 2 if y°-+-vy+1=0. 
The remarks in the text apply to all algebraical functions, explicit or implicit. 
