ana K ^ must =0, except tor tlie caseot &),, «■)«••• (O-j 



' ■' Yl Ts-- •'T'- ■' ' . . ; > ,.'-'^..'«;iiiron^ 



ifu'^z'"'^r being i*espectively identical with ^pA-g.i.i:;.} 7,v ^4'.'i»^,./ 



for wliich case R ' ^"* '' must be unity. I have gone through 



this bafculatiou and verified th^P^bMt')' {tfdM^i-46' etfect wliich.'^ 

 however, the following importaii^'^i^nei'fi'lizatioA of theoreih' iflj.^ 

 must be apprehended. '■ ', ,, . 



.r^S^r.^^nff^.'i^^'fl^S.rrT . V; .s ;.d.lo .di noqo /ooo-. 

 adJ 'to tijo-xi oj y,'"-*^ '•^^^' ^:^fti'(n'Uf'''^i^'VB (loofc f.R JiWiuqqB ai g'lad 

 pihf hfTR .«)nRrrrmT>t>!*i't" 6^Hvbd»,+-«-i!»vir) ^o gnoiifilo'i IfiJ/ti/ifi 

 and let (r) be any nuinber' less than («), and let any (r) -al-y combi- 

 nation of the 711 numbers 1,2,3 ... ??j be expressed by '^^,^^2•••*^?^" ., 

 where a goes through all the 'Mtifes MetnlediElt^ bfiirweeti (li) ^ 



and (it), /* bang , ' , , ,, ,. ,. 



o J.r^/)iTK,-Jj ful) ii^M^i^i'y:;.^i^ip^i'l4 . r>ilJo ur t Tx/o auo 



•"^ • • ■ 1 [' toif'io noJ8'(?! boJomJ^ 

 Then I say, that the compound determinant, ___ ___._..______-- 



^% • ^^e'^-'-^^Om ' ^»» + i ^m + 2 • . ^»i + » C29j . hg . . ^s^,^ j,„ + 1 Om + 2 • • V+ «_. 



T)70 bmrn% '1 7,vt: ..l'^^ ■^•X"^''x ^''^a"" ^^'^^V"'^ '^" uoiianBlqy _. 

 is equal to the following product, :hr)\ds.\?. orfi no 8jhf>mo'. 



.l.,;tnf^™l*'i•1wt-^rfO"^Mrf^^-H^ b'r)m*>'irpnox.tj^*fexioa adi ;tBdi — fcirli 

 '"^HS^iobv odJ vio'l-rrjilT TRifi fufi; .oiiihirHiiiijJ 'to iroiixjvylo ns gagnB-i 



».,...^-,^ff, yi.j^nL^. (»^-l)(/»-^) -ir: •.(?>f-^:-l<r,^) .t xroiJfi-^iJqO'iq lo 



ftd binow .t')0(dji.. 'i^l -y^ ",., v.;^ i!.-i(>'— l).M!.'ioqm3t adi dixv/ 

 andiot an ylqnnis fi (unUiniiU\/) IisjjJ )b ^ilT %j/oorio'i'i9 •jsdJasollK. 



ji'jbbxrg ^d b90^jbo ^^3>)(i»»»3)a(i-ji(afe'k»p)) uoxjpo8noo nl ,8V/' ^ ■ 



t.,.:'.,.,, i; i, ,,1, !,!■..•■ '■ ■■ <..:,:■ M'-H.,!-. 'h.' •v;m(^([- i: dtiv/ \J^J^'^\v^^ n;^''^^ ' 

 when r = l, we have the case already giy^it.ij^ ^|i^orein,j|_^,)^,aa;i|^ ,, 

 of course //." is to be taken unity. . , ,,.,;',. i., i;,.,,; - 



This very general theorem is itself several degrees removed 

 from my still unpublished Fundamental Theorem, whicl;i is a 

 theorem for the expansion^qf t|i6,,pypc|v|,Cj^^^jf idetei^inanljs. ,, ,,, ,, 



;., i^*-^^ Jw'lo fioiIxiXiBki/OK'aoijlqBJ'io J-ixuj kilnonr.'j iib dou u; 

 Ohs. The analogy upon which the extension of the Vander- 



