DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS 465 



Treating the rows of the new determinant in the same way as the columns of the 

 old were treated, we find 



I*, *1 = 



, 30 s *,,, 30*. 

 0**,, 40 s *,,,, 60**,, 

 50**, T , 



In the right-hand side a factor s can be taken from the first column, 0* from the 

 second, from the third ; and then from the first row, 1 from the second, s from 

 the third, giving as the power of the sum 



so that 



0, z}, = 0* [s, a;},, 

 or 



1 16. The result of the reduction of the ntic derivative is 



The method is similar to that used for the cubic derivative. Thus the numerical 

 factors which determine the algebraical multiples of the second, third, fourth, . . . 

 columns, to be added to the first in order to remove all differential coefficients of order 

 lower than d's/dxf, are respectively n 1, (n 1) (n 2), (n 1) (n 2) (n 3), . . . ; 

 the numerical factors which determine the algebraical multiples of the third, fourth, 

 fifth, . . . columns, to be added to the second in order to remove all differen- 

 tial coefficients of order lower than d*~ l 8/daf~ l , are respectively (2 1/1 !} (n 2), 

 {31/2! 11} (n 2) (n 3), {41/311!} (n 2) (n 3) (n 4), . . . ; the numerical 

 factors which determine the algebraical multiples of the fourth, fifth, sixth, . . . columns, 

 to be added to the third in order to remove all differential coefficient* of order 

 lower than &-**/daf-*, are respectively [3 1/2 1} (n - 3), {4 1/2 I 2 !} (n 3) (n 4), 

 { 5 1/3 1 2 1} (n 3) (n 4) (n 5), . . . ; the corresponding multipliers for the modifica- 

 tion of the fourth column are 



Jj (-*). jTij ( - 4) ( - 5), ^ (n - 4) ( - 5) ( - 6), ...; 

 and so on. 



MDCCCLXXXVIII. A. 3 O 



