DIFFERENTIAL EQUATIONS OF THE n"' ORDER. 



17 



whence, by putting successively r=l, 2, 3, ..., (n — 1) , all the desired 

 relations will appear. These are then the relations , that must be identi- 

 cally satisfied , in order that the values of the difterential coefficients of z 

 of the (n — 1)" order may render the equations (15) integrable. 



The equations (15) take a much simpler form, when the equations (30) 

 are n distinct integrals of the equation (12), that is: when they satisfy the 

 relations (28). In the equations (33) we may, namely, interchange the rows 

 or columns in any way, because then the determinants in their left members 

 can not undergo any change except, perhaps, with respect to their signs. 



Thus the equations (33) may be written 



Ar 



Ar 



Ar' 



,('■) 



ifr-i) 



r)(r-l) r>(r-l) r)(» 



T-)ir*n rio-.i) nc-'i) 



AC-) 

 1 /_A« — 1 



Î ■'-^n - 1 



nc«-i) r)(«-i) r)("-i) /ic-i) 







(34). 



Now choosing an arbitrary integer i such that Df'' ^ , the relations 

 (34) may have their left members multiplied by JDf ^ , that is : we may in the 

 determinants, represented by (34), multiply the terms in their first rows by 

 that quantity. The determinants in the equations, thus formed, will remain 

 unchanged, if from every term of their first rows we subtract the corre- 

 sponding terms of their second rows multiplied by the quantity Af* • Then, 

 in virtue of the equations (28), all the terms in the new first rows will be 

 severally zero except the first, so that (34) becomes 



Dr AV — D'o'A'r ,0,0. ...,0 -0.. (35) 



DV , £>?■', Dr, . . . , DV-, 



D', 



D[ , z>: 



D„^ 



J-'O 1 J-'l 1 -LJ-i , . . . , J-'n—X 



■'-'x 1 -'-'■i 1 • ■ • 1 -*^n-l 



T^rr+i) 



Nova Acta Reg. Soc. Se. Ups. Ser. III. 



1 ■'-'n-l 



