MR. W. SPOTTISWOODE ON DIFFERENTIAL RESOLVENTS. 229 



and differentiating, we may form the system 



- A^" + 2^a^a^ + (F - A')<2?' + . + E'^* + Y'a? + G' = o, "^ 

 . — Aa?a^ — . + Eo?' + F^* -\-Gce . = o, 

 Aa?' + , +E^* +r,^ +G=o, 

 aw^ + ^bx^ + ^cx-^- d =OjJ 



or, as it may be more concisely written, 



= 0, . (lo) 



M9) 



w" 



xx 



1 



X 



x-' 



x"- 



X 



I 



-A 



2E 



r-A' 



, 



E' 



F' 



G' 



, 



-A 



• 



E 



F 



G 



• 



, 



. 



-A 



. 



E 



F 



G 



• 



• 



• 



a 



3^ 



3^ 



d 



fi;om which we may eliminate linearly any three of the quan- 

 tities x^\ XX J x\ x'^y x^y Xy\, If we eliminate xx\ x^, x'^, we 

 have the differential resolvent, viz. * 



2E . E' 



-A E F 



. . E 



• « 3* 



. A^" + (F - AO<r' + F'.^ + G' 

 Ox 

 —Aa^ +F<2?+G 

 1cx-\-d 



= o;.(ii) 



the developed form of which is 



A'E«^" 

 + A{«(A'E-AE') -3aEF + 6Z>E^}^ 

 + {aA(E'F-EF') + 2aE(F^-EG)-6E"(^F ~cE)}<r 

 + {aA(E'G-EG04-2flEFG -2E^(36G-^E)} =o.. 



Hi2) 



This agrees with a result communicated to me by Mr. 

 Harley. 



It seems possible to exhibit the resolvent as a single 

 determinant; but as this is of the i6th degree, and does 

 not (at least so far as I have found) exhibit the discriminant 

 as a factor, I have set it aside as top unwieldy for use. 



