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Mathemathics. — “On a class of differential equations of the first 
order and the first degree.” By Prof. W. Karren. 
1. In the last meeting of this Academy Prof. J. pr Vries gave a 
geometrical criterion for determining whether or not a given differen- 
tial equation of the first order and the first degree may be reduced 
by a homographic substitution to a linear equation or to an equation 
of the form 
dy _ Ney +P@y+ Qo) 
da R(w)y + S(z) 
The object of this paper is to examine the general form of all 
those equations which by a homographic substitution may be 
reduced to the equation (1). It is evident that this general form will 
give at the same time all the equations which are reducible either 
to the general equation of Riccati, or to the linear form. 
(1) 
2. Let the substitution be 
__ autayta, a ___ biutbvdb, B 9 
me c,ute,vte, eae Pine ae cutevte  Y @) 
where a,b,c are constants, then the equation (4) is 
dv CBN HBP H7°Q']—Ar [BR +78] 4 
du y B[BR*+-yS"]—D[BP?N* + ByP*+y’Q'] 
where 
A=b,y—o,8 C=a,y—¢,a@ 
B= bir 0, 6 Dap oa 
and 
var(s) ace ¢ =0(<) r= R(S) s=s(<) 
Y Y 7 7 Y 
Transforming now to parallel axes, taking as the new origin o 
coordinates the point where the lines «== 0 and y=O meet, we 
find the new equation by substituting 
EA ke 
RA 
(a,c,) 
= (4,¢,) 
(4,0;) == Gicg — Oren eae 
+u,v 
In this way, we get 
a=au+av',B=bv+),7 +e=8 Hes, yoouuvt+e,v' 
e being a constant, and 
55 
Proceedings Royal Acad. Amsterdam. Vol. XI. 
