SATISFYING LINEAR DIFFERENTIA!, EQUATIONS. 19 



In order to proceed to the determination of the second column it may be noticed at 

 once that the coefficient of y r l in any equation is exactly equal to that of x r l in the 

 corresponding equation from the first column with o-j and cr 2 interchanged, which 

 includes the interchange of l p - k and 6 2 p - k . In the equation 'Y, i+p _ l , therefore, the 

 coefficient of y^ is identically zero, while the unknown 6-Js are now all determined. 

 The closer consideration of this equation is deferred for a moment. 



The coefficient of y in Y, i+p = coefficient of x* in X, i+p with o^ and <r 2 interchanged 



= (<r a <7i)(er a o- 8 )...(crg o-.,) i= 0. 



Similarly the coefficient of y% in Y n+p+1 , and in general of y k l in Y + p + k _-i, is not 

 zero, while Y ei+p+i _! does not contain any element y m * for which m>k. 



For the third column the equation Z ei+p _ 3 determines 0^, and the two equations 

 following, Z, i+p _ 2 , Z. i+ p_!, are still independent of z^, z 2 1 ..., while the equation 

 Z.j+p+A-! contains z^...z^, the coefficient of z k being &(cr 3 o-i)(o- 3 cr 2 )(cr 3 o-.,) 



In the same way, if the elements of the e t ih column be denoted by and the 

 associated equations by ft, the equation l p determines 0^', ft ;j+ i...ft ;)+6i _ 1 do not contain 

 fi 1 ..., and ft,+,, contains & 1 only. 



11. So far the matrices XP---XI nave been taken to be simply diagonals. It will now 

 be shown that the insertion of constants to the right of the diagonal in the first e v 

 columns of XP can be carried out in such a manner as to affect none of the conclusions 

 hitherto made, while they may be chosen so that the equations Y^^, Z, i+p _ 2 , 

 Z.^p-j, &c., are all satisfied. 



Denoting by a,-,- a constant in the i ih column and _/ th row, i>y, i^f\, ,/<fi, the 

 Y equations become 



(0) (a ?)+1 -^ 41 )7/ -.r,,=0. 



(1) (a p+1 ~0 p ^)y l + (a p -0 p )y a -x l -a !>l x =0, 



(r+l) (a p+l -0 p+l )^ r+1 + (a p -0 p )>/ r ...-.r r+l -a.-\r r = 0. 



These equations are to be treated just as they were before the constants a were 

 introduced the same elements remain undetermined as before, but at each stage the 

 quantities found presumably contain a 21 . 



We see at once from equation (1) that the coefficient of 21 in y^ (the first element 

 being excepted) is simply y . In fact, it can be shown step by step that the coefficient 

 of a 21 in y r+l (the first element always excepted) is exactly that part of y r which is 

 independent of a zl with 0^ increased by 1 ; and therefore the coefficient of a 21 in Y r+1 , 

 when the values ofy r 2 ... as far as they are known are substituted, is equal to that 

 part of the left-hand member of Y r which is independent of a 21 , #/ being increased by 

 unity. 



Now Y! is independent of a 21 and of 0^ and vanishes, thus Y 2 is independent of a 21 , 



D 2 



