158 



again emphasize the incoinpleteness 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 equation. This is the equation dealt with in 

 the paper. It can be written in the form 



2z 



i=l i 



%, Xi, X2, 



,Xn 



%, Xi, X2, . . . . , Xn 



The restrictions made on this equation are that all common factors have 

 been removed from -o-, A-j, A.2, .... -A.„; that there is also a set of values 

 of the variables ^, Xj, Xo, . . . . , Xn in the vicinity of which the func- 

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



Forsyth, in his treatise on Partial 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 determined and a new and com- 

 plete classification is given of all the integrals of the equation. 



