( 815 ) 
3. Now we will show that where a differential equation of this 
form (4) is given, there always exists a homographic substitution by 
which this equation may be reduced to the form (1). 
For let 
then we have 
Kee 1 
Hi wek, (=. zerk CE; 2) 
ane | y 
me Ee 1 
M, = M, (uv) = M, y ’ y ae” ea ‚) 
etc. Thus (4) reduces to 
dy tA, (1,2) + ey’ +-L,(1,0)y +N, (1,2) (5) 
de {#H,(1,x)—K,(1,2)y+e2L,(1,2)—M,(1,c) © ° ~ 
which is of the same form as the differential equation (1). 
4. Therefore we have proved this: 
Theorem. The necessary and sufficient condition that a differential 
equation of the first order and the first degree, having a singular 
point in the origin of coordinates, may be reduced by a homographic 
substitution to an equation (1) is that it may be written in the form 
dy  K,+M,+y(N.+9 6) 
de BTL, Fang 
Corollary 1. The necessary and sufficient condition that a diffe- 
rential equation of the same kind may be reduced by a homographic 
substitution to an equation of Rrccarr is that it has the form 
dy _ M,+yN,+0) : 
pee aay 
Corollary 2. The necessary and sufficient condition that a diffe- 
rential equation of the same kind may be reducible by a homographic 
substitution to a linear equation is that it has the form 
dy _M,+yN, (8) 
Mae DeL ANT. naj REE 
5. With respect to the equation (8) we may remark that it is 
equivalent with 
dy deren M, +4 N, 
desen Leste N, 
as the numerator and the denominator of the second member may 
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