158 



again emphasize the incompleteness of the existing theory of partial differ- 

 ential equations of the first order. 



In attacking this problem the logical place to begin is with the simplest 

 case, namely, with the linear ecjuation. This is the eciuation dealt with in 

 the paper. It can be written in the form 



2z 



i=l i 



^. Xi, X2, 



,x„ 



=z 



*, Xi, X2, 



x„ 



The restrictions made on tliis equation are tiuit all conmion factors have 

 been removed from -o-, ^i, A.2, . . . , ^n] that there is also a set of values 

 of the variables ft, Xj, X2, . . . . , Xn in the vicinity of which the func- 

 tions -A^i and -o have no branch points and otherwise behave regularly. 



Forsyth, in his treatise on ParlUd Differential Equations published in 

 1906 goes to much labor to give solutions that are examples of the so-called 

 special integrals. In the present paper a means is developed by which all 

 the elusive special integrals can be readily determincci and a new and com- 

 plete classification is given of all tlu* integrals of the ocpiation. 



