1 >> f ^^ ' uepresent : auxb^xey. 





[D-fifiq arii 'io 

 MIT 10 8Jat»ioift90o ofil to Juo vfavii/ii y/':, , ' ^ Ifcujlxim flli'// 

 ifiii :j;i'jmjn9 ■ii59lo « Jiifl) •i9mo nl . .': '^ ^to f-noitsup9 



j>!i:i'nq oi 7-XB8H309II «i ;ti ^o[6ie^oq TT'ftP x ^«t x cy iyio'^dl 9ill io 

 l>^oiit s ^9tu oJ bavo-iq gBxf 3-jnohy(fr- «« x i7X<;/9;!ifia Ixid w^a « 

 lla oJ sidfiDilqqjj noiJBJoa 'io boilisiiL^-x.iiif^o.y^ i'il(!'''P^ Jjj>i97/oq 



Aitnougli not necessary tor our nnmcdiate object^ it may not 



be inopportune to observe how readily this notation lends itself 



to a further natural extension of its application," '.'""'"r^' 'V . V"''" 

 ^^ boionoh so ia-%im 



{> will natm'ally denote , , » 



ab cd ab c(l 

 a.^ jB <y8 a./3 



j '{accxbm^f {cjxdS)\_( {ayxbSy 

 l-ia^xbc^j X-^Sxdy)J \-{a8xl^ 



bB)] j {cu.dl3y 

 And in general the compound determinant noisd ?a YJijs'f 



{«! 6, . . . /[ Oo bc^ . . . h ... o,. b^ . . . l^ ~\~ 

 a^ yS, . . . Xi «2 /Sg . . . Xo «,. i3^ . . . \-J 



will denote 



where, as before, we have the disjunctive equiatibri ,^ ' .iT 'j'l 

 ^u vi> • • • V9.=;lj ^j^nw5t9r|o 'to 'rorr vtrtnBrfp "io 



As an exam])le of the power of this notation, I will content 

 myself with stating the following remarkable theorem in com- 

 pound dctciTTiinants, one of the most prolific in results of any 

 \vith which I am acquainted, but which is derivpd from a more 

 particular case of another vastly more general. TJie theorem is 

 contained in the annexed equation. iiri i i: " 



{«, (t^ . . . a, «,+ i <7, Uc, . .. Ur (tr^2 • • • «! «2 • . . (',■ (lr + s\ 



«j «2 . . . ftr «r + l «! etj ■ . . «r «r + a «I «.2 ...«,• Uf + aJ 



_ r fl, «2 . . . ftrl '"' ^ r«i fl^ • • . «,• ('r+l (tr+2 - • • «r+« \ , 



L«, a.2 . . . UrJ ^-«i «2 . . . a,- (S^r+l «r+2 •••«(•+,? -* 



It is obvious, that, without the aid of ray system of imibral or 

 bilitcral notation, this im])ortant theorem could not be made the 

 P/ii/. Mmj. S. 1. Vol. 1 . Xo. 4. April 1851 . X 



