14 ALF GULDBERG. M.-N. Kl. 
equations which, combined with (7), are eqvivalent to the three systems 
obtained above.! 
We could also have obtained the same equations in the following manner: 
Of the four equations: 
Aa + BB+ Gy + Dö+ H=0 
dr — adı — Bdy =0 
ds — pdx — ydy = 0 
dt — ydx — ddy = 0, 
only three are independent. But the last three equations are obviously | 
distinct. It is then necessary that the first equation is a linear combination 
of the last three, and we have 
Aa + BB + CG + DÖ + H=X (dr — adx — Bdy) + Y(ds — pdx — 
— ydy) + Z (dt — ydx — ddy), 
X, Y, Z being independent of a, 8, y, 6. We have further, 
A+ Xdx = 0 
B+ Xdy + Yax = 0 
C+ Ydy + Zdx=o ¢ (14) 
D + Zdy = 0 
H— Xdr — Yds — Zdt= 0 
Eliminating X, Y, Z between these equations, that we find again that 
Ady — Bdy?dz + Cdydx? — Ddx* = 0 
Adxdydr + Bdxdyds — Adx*ds + Ddx?dt + Hdx*dy = 0? 
5. We have seen that the knowledge of two integrals, # — 4, 
v = 4, of one of the three different systems of the form (13), enables us to 
construct a general first integral, 
D—=u—f(v) = 0, 
of the partial differential equation (3). And the solution of this partial 
differential equation of the second order would lead us to the final 
solution. 
1 cfr. M. Falk: »On the integration of partial differential equations of the mth ordere, 
Nova Acta Regie Ups. 1872. = 
2 Without entering here into details, we may remark that the equations (14) give a very 
convenient form for discussing the various combinations of total equations, which 
arise when some of the coefficients of the given equation (3) vanish, 
