1897.] Differential Equation of the First Order. 



285 



The right-hand side of this equation vanishes if the equations 



(6) and (7) are satisfied, m being zero or any whole number up to 



n inclusive. Thus in general the equations (6) and (7) give p r as 



dz 

 the value of ^— and therefore afford a solution of the differential 

 ox r 



equation (1), which was to be proved. 



The relations (6) with f= form a ' complete system,' the 

 singular solution of which is given by (7). Thus we may say that 

 all the solutions of /"= are singular solutions of complete systems 

 to which /=0 belongs. It is not necessary to form all such com- 

 plete systems in order to have all solutions. 



.(8). 



Gases of failure. 

 % 8. The proof will fail if 



d(f,lh, ... Mn) ^ Q 



d(z,p lt ...p^ 



If we multiply the equations 



(f> Wi) = 0, etc., (wj, Wo) = 0, etc. 



by the second minors of this Jacobian, formed by leaving out the 

 derivatives with respect to z and p r , and add, we have, as other 

 forms of the relation 



d(f, t^, ... u n ) 

 d(z,p 1} ...p n ) 



= 0, 



that 



d(f Mj, ... u n ) 



= 0, (r=l,2...*i) 



d(w r ,p 1 ,...p n ) 



Hence it follows that all the determinants of the matrix 



.(9). 



.(10) 



