an 
Mathematics. — “On a differential equation of Scurärm.” By 
Prof. J. C. KiLuyver. 
As a suitable example of the method of solution due to Prarr 
ScurÄrLt has determined the general integral of the equation 
a, (",P;—*,P.)” + a, (@,P,—2,p,)" + 4, (epo) = 1 
(Annali di matematica pura ed applicata, serie 2, t. II, p. 89—96) 
and in his Theorie der partiellen Differentialgleichungen Mansion 
has repeated the calculation of ScanArni. As Mansion remarks this 
treatment of the equation does not allow to maintain the symmetry 
with respect to the variables; therefore we will show in the fol- 
lowing lines that it is possible to obtain the complete integral of 
the equation with preservation of the symmetry by means of JAcosr’s 
method. 
By putting 
EP — &.P, = A,, 
FP om te 3 A,, 
Pin Mal A, 
the given equation passes into 
FS 0A GAY 4 ag Aa 0; 
The system of simultaneous differential equations to be considered 
here becomes 
da, >> dp, 
av, ar, dj eee a,p,A,—asp,A, Ter 
One derives from it immediately 
da, Et dA, 2A,dA,  p,dy, 
cee ee ame A (a NAA ioe 0 oe tee des: 
This furnishes two integral equations 
eas LP + ps —m* =0, 
Fe ee AL A eg, 
The two functions f, and /, are in involution. For we have 
LA Py | = 0, LA, Pad == 4 AP sPs) LA, *p"] =-4 A, p‚ps- 
From this ensues 
[Ar Zp] = 0 
and furthermore also 
eae fi] == 0. 
So one has to solve the partial derivatives p,, p,, p, out of the 
three equations 
Pet ON DEE aa 
and to integrate afterwards the differential equation 
dz = 2pide,: 
A direct solution of p,, p,, p,; cannot be given. Therefore we 
