38 ALF GULDBERG, M.-N. Kl. 
— mG = — mG 5 — — (nGp + mGg) 2 — — (nGr LaCie D — 
— (nGs + moi 2 a+ (nD + mF + FS + GE ps 
+ (E+ mA + BE) = o, 
Hence we have the auxiliary equations 
ax dy dz ap 
—1G —mG —(nGp+mGq) —{(nGr + mGs) 
— (nGs + mGt) — nD+mF+F_ G mE+mA+E 
Eliminating # and z from these equations, we have 
Gdr + Ddz + uFdy — Fds =0 
Gdt + 1Edx + Ady — Eds = 0 
dp — rdx — sdy = 0 
dq — sdx —tdy = 0 
da — pdx — gdy =0. 
This then is the system of total differential equations upon the inte- 
gration of which the determination of # will depend. 
If Afand w receive their different values, determined by the quadratics 
(A) and (u), we get different systems for determining z. If from any of 
these systems we can deduce two integrals of the forms 
u=av=b, 
it is obvious, from what precedes, that 
u=/(v) 
will constitute a first integral of the proposed equation (I). 
