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On THE REDUCTION OF IRRATIONAL ALGEBRAIC INTEGRALS TO RATIONAL ALGE- 
BRAIC INTEGRALS. By Jonn B. FAuacut. 
The Inverse operations of Analysis are more interesting and fruitful than the 
Direct, since each demands a new field of quantity or a new kind of function in 
order that it may be possible without exception. Thus negative numbers have 
their origin in subtraction, fractions grow out of division, and irrational and im- 
aginary numbers arise in the extraction of roots. The same thing is true of inte- 
gration considered as the inverse of differentiation. 
At the time of the discovery of the Calculus the algebraic and certain ele- 
mentary transcendental functions were known. The algebraic functions included 
all those expressions which can be formed by a finite combination of the processes 
of addition, subtraction, multiplication, division, involution and the extraction 
of roots. The transcendental functions included the exponential, logarithmic, 
trigonometric and circular functions. These will be called the elementary fune- 
tions, and exclude the infinite series. 
It is a fundamental theorem of the Integral Calculus that the integral of any 
rational algebraic function can be expressed in terms of the elementary functions. 
This is sometimes expressed by saying that any rational algebraic function can be 
integrated. 
The attention of mathematicians was early directed to those integrals that are 
made irrational by the presence of the square or other root of a polynomial of the 
first, second and higher degrees. It was soon found that if the irrationality was 
due to a square root of a polynomial of the first or second degree the integral 
could be expressed in terms of the elementary function. The integration being 
accomplished in each case by reducing the irrational function to a rational func- 
tion and then performing the integration. This method, however, was found to 
fail, in general, as soon as the polynomial under the radical is of the third or 
higher degree. The investigation of irrational algebraic integrals led to the dis- 
covery of the Abelian functions of which the Hyperelliptic and Elliptic functions 
are special cases. By means of these functions the integral of any algebraic func- 
tion can be expressed. 
The integrals under consideration here are known as Abelian Integrals and 
are defined thus: 
{F (x, y) dx. 
where y is defined by: 
fu (x, y) = 9, 
and F denotes a rational function of x and y. 
a a 
