20 M. Falk, On the Integration of partial 



we obtain by its integration the general solution of (12), as each of the first 

 integrals involved one arbitrary function. 



2) If, however, the equation (20) has equal roots, then we find a 

 less number than n distinct first integrals. If from these integrals we 

 can obtain some of the differential coëtficients of z of the [n — 1)" order 

 and after their substitution in some of the equations (40) integrate them and 

 so go on similarly as in the former case, the solution of (12) will be de- 

 pendent on the solution of another partial ditferential equation of an order 

 lower by one or more units than the given equation. If this new equation 

 be linear, we may anew apply the given method to integrate it; if it be 

 not linear, the method of the §. 8 of this memoir must be used to obtain 

 its solution, which will in that case usually be a complete or a mixed one. 



If the equation (20) cannot be completely solved, but some of its 

 roots are known, the same method, as in the latter case (when (20) has 

 equal roots), must be applied. 



Sometimes we may by artifices reduce the latter to the former case 

 by taking instead of (12) a new equation, such that its equation (20) has 

 no equal root, and at the same time such, that its roots become identical 

 with the roots of the equation (20), belonging to (12), for certain values of 

 some constants involved in the new assumed equation. In order then to ob- 

 tain the general solution of (12) from that of the new equation, the method 

 of limits is to be applied. 



The equations (11), or those between (10) and (11), shew, that the 

 (p and "4/, which belong to a certain first integral, are functions only of 

 each other, if we consider all quantities, that they involve, except the dif- 

 ferential coëfl:'icients of z of the (u — 1)" order, as constants; tliat is to 

 say: if between (p = a and -J/ = ß we eliminate any of these ditferen- 

 tial coefficients, all the others will disappear at the same time. We may 

 also express that property briefly thus: <p and ^' are functions of one and 

 the same function of the ditferential coëtficients af z of the [n — 1)" order 

 and involve besides none of them in any other w^ay. This diminishes the 

 labour of solving tlie equation <p = /(\^) with respect to the function in 

 question, which is necessary for obtaining the expressions for the ditferen- 

 tial coëtficients of ;: of the (n — 1)" order; it deserves, however, attention, 

 that in general it is impossible to solve the n first integrals with respect 

 to these ditferential coefficients, provided these difl". coëff. really are in- 

 volved both in (p and \|/. But, in this case, we must seek the complete 

 primitive of (12) and may then, if the functions (p really involve tlic diffe- 



