12 ALF GULDBERG, M.-N. Kl. 
The linear system which we get by equating to zero the last two 
factors of the first two equations of (11) combined with the third equation 
must be rejected as irrelevant. 
4. Asa result of the foregoing investigations, it is found that the func- 
tion z is to be determined by the solution of three simultaneous linear partial 
differential equations with eig4t independent variables. Now the theory — 
of linear systems shows that the number of integrals of such a system 
cannot exceed five. This theory enables us both to determine what the 
number of integrals is, and to construct the system of total differential 
equations upon which their discovery depends. We will here consider 
this last problem. 
To reduce the determination of the first integrals of (3) to the solu- 
tion of a system of total differential equations. 
Each of the systems (12), (12°), and (12°) presents # as satisfying simul- 
taneously three linear partial differential equations of the first order. In 
the following investigation, we consider # as given by the system (12“). 
To deduce the value of # thus conditioned, it will obviously suffice to 
multiply the second of the partial differential equations (12) by an in- 
determinate multiplier #, and the third equation by a multiplier z, and 
add these results to the first equation, so as to form a new equation 
which will be linear and of the first order, and which, on account of the 
indeterminate quantities # and x, will be equivalent to the three equations. 
If in this way we combine the equations of system (12“), we have 
ou 
eu Ou ou 
Du Ds + DP — De gg + (Dr — DA G+ 
ou 2, ou ou 
+ (Ds — i, Di åg + (mD + nå,) 37 + nd, = + 
ou 
+ (å, Am + 4 + 1H) = 0 
Hence we have the auxiliary equations 
EE LE ee 
D —i,D” Dp—1,Dp Dr—i,Ds Ds —i, Dt 
dr ds dt 3 
~~ mD + nh; na, MAmtn-+ A, 
