OP LINEAR DIFFERENTIAL EQUATIONS. 241 



* — (2'5 . 54. 7 . 11-^2'°. 3^ .3527cr 



+ 2' . 282949^* — 2 . 1x7890?^ 



+ 2^ 3.757% 

 — (2" . 1801—2^ . 150757 



+ 71 0357* — 2* . 3 . 757^)57} ; 

 whence the resolvent 



5V(4'-^)g + 5V(2^3*-7^)g 



+ 3.5'. i7«^(2' . 5-^)^ + 2 . 3 • 5'(5*-3*^)^ 

 + 2.3^=2.3. 



11. Collecting results and passing, as before, from the 

 ordinary to the Boolian or symbolical form, we find that 



For the quadratic, the resolvent is 



For the cubic, it is 



3^[2D]^2^-2(3D-4)(3D-2)6^2/=MV. 

 For the biquadratic, it is 



4' [3l>] 'y - 3 (4D - 5) [4l> - 2] Vy = [3] '6". 

 For the quintic, it is 



S*[4D]*y-4(5D-6)[5D-2]V2/=[4]V. 



From these four cases the general form for the resolvent 

 of the equation (II) is sufficiently obvious ; but I have 

 thought it well to test that form by the case n=6, or, what 

 is the same thing, by the sextic equation 



y^ — 6y^ + 557 = o, 



for which the general form (C) gives 



6' [5D] 52, - 5 (6D - 7) [6D - 2] Vj, = [5] '6,. 



12. Returning by the usual method from the symbolical 



SER. III. VOL. II. R 



7 



