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Mathematics. — “On some special cases of Monen’s differential 
equation”. By Prof. W. KAPTEYN. 
If as in general p, q, r, s, ¢ represent the first and second differ- 
ential coefficients of a function z of two independent variables «x 
and y, the differential equation appearing in the title is 
Hr + 2Ks + Lti+ M=0, 
where H, K, L, M are functions of z, y, z, p and g. For the solu- 
tion of this differential equation MoncE has given a method based 
on the determination of two intermediate integrals of the form 
u=f(v), 
where wv and v are dependent on «, y, ze, p and g and f represents 
an arbitrary function. This method is however deficient, the existence 
of intermediate integrals depending on certain relations between the 
functions H, K, L and M, thus far unknown. 
For the purpose of making this method more practical, { have 
tried to trace these unknown relaticns. However, these relations are 
very intricate; for this reason I have confined myself for the present 
to the simple cases where MonGsk’s equation consists of two terms 
only. For these cases I have succeeded in finding the necessary and 
sufficient conditions which the only remaining function must satisfy, 
if we suppose that the equation has two intermediate integrals. 
From these conditions I have then deduced the most general forms 
for the only remaining function and determined the two intermediate 
integrals belonging to it. 
The result of this investigation is the following, as for as 
concerns the most general forms of the function and the two 
corresponding integrals, f, 9, w representing everywhere arbitrary 
functions of the arguments indicated when necessary between 
brackets. 
ih rr — AS = 0. 
In this case the form is 
À Es E ») 
q — 9 (y.p) 
and the intermediate integrals are 
