DIFFERENTIAL EQUATIONS OF THE n"' ORDER. 13 



Denoting -^ by m we get, after the quantities /* have been eliminated, 

 for determining m the equation 



S" (—1)' U, m"-' = (20), 



which togetlier with tlie equation 



dij — mdx = (21) 



are equivalent to (17) and (18). 



In oi'der to be able to use (19), we mnst determine the quantities 

 jM, from (17) and (18), or else, putting U„f/,^= TF, , we have to determine 

 if] the equations 



T^„= Z7„, JV,-{- W,^, m=U, (i=l, 2, ... , n-1) (22). 



(We need not necessarily use the equ. TT''„_, m = U„ , which follows 

 from (17)). Having calculated the quantities PT, the equation (19) maybe 

 used in the form 



S' ^V.dz,._,_,„ == Vdx (23). 



Now the equations (20), (21), (22) and (23) supply the place of 

 (17), (18) and (19); and by integrating the former group of equations, the 

 functions f/ and (/^ will be known. 



Suppose m a root of the equation (20); we substitute this root in 

 (22) and get a system of quantities W . Then (21) and (23) will be two 

 ordinary simultaneous differential equations, where the coefficients of the dif- 

 ferentials are known functions of œ , y , z and the differential coefficients of 

 z up to the {n — 1)" order. Integrating these we get a system of values 

 of (f and ip. And in this manner there will be as many such systems, as 

 the equation (20) has distinct roots, supposing, of course, that the integra- 

 tions are always possible. If, on the contrary, the equation (20) has equal 

 roots , the number of these systems Avill be less than n . As now an arbi- 

 trary relation between a ff and a 0, constitnting together such a system, 

 is a first integral of (12), there will be found as many first integrals of 

 the given partial difterential equation , as the equation (20) has distinct roots , 

 all the integrations in (21) and (23) being supposed possible. 



