3 
order of the convertent equation, and the possibility of the 
reduction of order depends upon the circumstance that 
1 dd 
(1 + 0 2 ) m dv 
is always integrable in terms of algebraical and trigonome- 
trical functions. When m, which here represents 2n — 1, is 
greater than unity, the case of m may be made to depend 
upon that of m — 1, and so on. 
Again, since 
:= e + 
Ki 
l + e 2 " ' \ + id + \- id ' ' 
where i is an unreal square root of unity we see that the 
conversion of the integral of (10) may be made to depend 
upon that of the integrals of two other proper fractions, 
whereof the denominators are of the n - th degree only. 
For 0 is supposed to be rational, and consequently imme- 
diately integrable. It seems therefore that in many cases, 
and perhaps universally, the conversion of the integral of a 
proper fraction may be made to depend upon the solution 
of a linear differential equation of the first order. For if 
one and only one particular integral of a certain linear dif- 
ferential equation is a linear function of one and only one 
particular integral of a second linear differential equation, 
and also of one and only one particular integral of a third 
linear differential equation, then each of the three particular 
integrals may be assigned by means of a linear differential 
equation of the first order. This is shown as follows. 
From the first equation eliminate its dependent variable by 
means of the given linear relation, and call the result the 
fourth equation. Then eliminate the dependent variable 
between the second and fourth equations, and the result will 
be a linear differential equation which will in general have 
one, and only one, particular integral in common with the 
third equation. Hence this one integral can be found by 
means of a linear differential equation of the first order. 
(13) 
