157 



(Abstract.) 



On the General Solution and So- Called Special 

 Solutions of Linear Non Homogeneous Par- 

 tial Differential Equations. 



L. L. Steimley. 



The integrals of a partial differential equation of the first order were 

 first classified by Lagrange, who separated them into three groups, namely, 

 the general, the complete, and the singular integrals. For a long time this 

 classification was thought to be complete. In fact, Forsyth in his Differen- 

 tial Equations, published first in 1885, gives a supposed proof of a theorem 

 stating that every solution of such a differential equation is included in one 

 or other of the three classes named. This error is also carried through the 

 second and third English editions and the two German editions, the last one 

 being published in 1912. 



In 1891 Goursat pointed out in his Equations aux derivees partielles du 

 premier ordre, that solutions exist which do not belong to any of these three 

 classes and showed indeed that the existing theory was not complete even 

 for the simplest forms. 



In November, 1906, Forsyth, in his presidential address to the London 

 Mathematical Society, emphasized the fact that the theory is incomplete, 

 and in his closing remark says: "It appears to me that there is a very defi- 

 nite need for a re-examination and a revision of the accepted classification 

 of integrals of equations even of the first order; in the usual establishment of 

 the familiar results, too much attention is paid to unspecified form, and too 

 little attention is paid to organic character, alike of the equations and of the 

 integrals. Also, it appears to me possible that, at least for some classes of 

 equations, these special integrals may emerge as degenerate form of some 

 semi-general kinds of integrals; but it is even more important that methods 

 should be devised for the discovery of these elusive special integrals." 



Forsyth also in an address delivered by request, at the 4th International 

 Congress of Mathematicians, takes advantage of the opportunity' offered, to 



