DIFFERENTIAL EQUATIONS OF THE «"' OrDER. 37 



together Avith what (61) becomes on, when n is pnt = 2. Here m and 

 jbt, are the two roots of the equation 



Co"*'-C,,'H + C,. = (67); 



and, as either of these roots may be used as the vahie of m, the other 

 being that of ^, tliere will be two auxiliary systems. Now one integral of 

 each of these systems together with the given equation will in general give 

 the values of Z2,û, ^i,i and z^^, and these values are then to be sub- 

 stituted in 



dz^^ = ^2, dx -\- Ci, 1 dy , d:o, i = ~i, i dœ + ^o, 2 dr/ , 



after the integration of which we get expressions for c, „ and ~-o.ii that 

 render the equation d: = :^ ^,dx-\- :„idy integrable. Tins it is easy 

 to see, because, independently of the values of a, and /3, we get, when 

 the roots of (59) are all unequal, (h -(- 1) distinct first integrals of the 

 linear equation (55), that render the equations (61) integrable, and among 



these first integrals there will be one (belonging to the root m = -j , 

 which is a relation involving (54) as a special case, viz 



-t (X 1 y 1 2 , *! , ) ^0,17 • • ■ 1 ~ n , ) ■ • • ■> ~0. ti) = ■^'^ 1 



A being an arbitrary constant. Putting then ^1 = 0, the operation 

 is restricted to the investigation of a (complete) primitive of the given equa- 

 tion (54), instead of solving the more general problem to seek a primitive 

 of (55). The arbitrary quantities a. and /3 may, of course, be determined 

 in the manner before mentioned. 



If (64) be linear with respect to the differential coefficients of : of 

 the second order, we may from (Gl) and (65) get an auxiliary system, 

 that does not involve these differential coefficients, because Co ' C 1 ^^^^ 

 r„ 2 are then independent of them. Thus if (64) be 



Rr + Ss + Tt — V ^ (68) , 



where r = ^2,01 « = "i,i» ^ = '0,2» (67) becomes 



Rm'—Sm+T=0 (69), 



whence 



