^06 '.«'yMr. J.^J. Syk<istev on the Relation between fke-^\ - \f 



ttat the' gj'zygetic connexion between the minor determinaBtS of 

 U and V of the same order is linear, as has been ah-eady aiitici- 

 patively announced. 



The problem which I have treated above is only a particidar 

 case of a more general one, T\'hich may be stated as follows : 

 given V={aiX^+a^.Xo+ . . ,a^.a,\f,'snCi supposing m linear 

 equations to be instituted between .r„ x^ . . . .y„, so that U may 

 be made a function of {n — m) letters only, to express any minor 

 determinant of the reduced form of U without performing the 

 process of eliminatic^n between the given equations. Let the 



givpn pquptioia^jbe writt^^'im ':;i[^^;; jii;;';; 



^^" n n V 4-fr n -»• 4L ' -4- /T ff ' , » * — 0"'*"^*^*' ^'^* 



(flOli-iM-Mli 1! 111.'. ii<'!j>Miiji |ii j;-i;-) -n;!)!';' In^'i -i'l I Mi'l i_)')JtM80*K{ 



; snoi/do liftijil ifKnt-iJsl <i -itBi'. dt <i^..')(nii;t fc. Tcl'itiu L,'i3V97/od 



and let it be converied (ii^-^liich takes nothing away from the ge^ 

 nerality of these equations) that «„+,..<?,.+., shall signify zero for 

 all values of /• and s cuncurrenfhj greater than zero. Suppose 

 tbat i?^^ la?^ . .! . ,T,„j being eUminated, U becomes of the form!;. •, 



"lit ri'i -I't ' /ii '; '■ ;^ i,i ■■ I 7, „ j1 'i' ,h ~, \t . ■■■ 71*1 'loj 



., lO»iJ.l.X,„ + i tW,„ + 2-<<wi+2 -r • • • -T'Jn-^'-n) s ^ / r 



-(;o J in. TMpi' "/If III ';,. ji .■>::! rii ;lil!i;! ■!;! ■. ■ ' ■.'■ • '• • )''''|i!> - 



and suppose tbat we wish to determine the.value t>f the complete 

 determinant of this last function, the ianswei* will be found M be 



y^Hj+V ^^+.2 • • ■ ^«\U- J ."'j ^2 • • • ^n 4«+i • • • fn-^m ,1 . ^ 



•iBlimfr; Y-rr//^^j)iaififiji;jJ<i^jiii'\fe'jV()Iqnia o/fid I fl)idv/ xioiijB^toa 

 botniBn[crfrlriin'VMr ivcff 1 ff'"iif///»iiliv^ ^o&norcrrobfffiY 1o .tBril oJ 



the squared division being, a^ is obvions>''a function ibhlyf off ftbe 

 coefficients of the transforming ec[uations, and depending for its 

 value upon the })articular (m) quantities selected for' elimination. 

 The dividend, on the contrary, is independent of this selection, 

 but involves the coefficients of the function combined with the 

 coefficients of transformation. Tliis is the symbolical represen- 

 tation of the theorem given by me in the postscript to my paper 

 in the Cambridge and MathematicalJournal for November 1850. 

 Suppose, now, more generally that we wish to find any minor 

 determixiant. The solution is giveni by the equation; ,i\>.ih ot m\ 



{J „ t. K - - T I ift'ifirifliii; 



OOm+l Od„, + 2 . . . DOm+s I 

 O0m + 1 0<Pm + 2 • ' ' tf<pm+s-^ 



(wherein the two groups ^„,+i,!!l?,„+2',l'..fc,; '^m^\y^m+i^/-4m.^s 

 are each of them (s) cUfFeiing, or wholly or in part agreeing indi- 



