ON THE THEORY OF LINEAR TRANSFORMATIONS. 
(5r = apbgkn — boalhm — cndiep + dmcjfo — elfocj + fkpied + gjhmla — ignbk 
— ihjocf + gjidep + kflamh — lekbng + mdnbkg — ncmhal — obpied + paofjc 
+ apbglm — boahkn — cndejo + dmcfip — elcfip + fkedoj + gjhakn — hibgml 
— ihjode + gijpcf + kflrnbg — leknah + mdknah — ncmlbg — obpicf + paojde 
4- apcflg — bojehk — cnjehk + dmilgf — elmpbc + fkadno 4- gjadno — hipmcb 
— ihadno + gjbmcp + kfcbpm — eladno 4- mdjehk — ncilfg — obilfg + pahejk 
+ apcflm — bodkne — cnahjo + dmbgip — elbgip + fkahjo + gjednk — hicfml 
— ihknde + mdahjo + kfipbg — bocfrnl + gjcfml — ncpigb — leahjo + paknde 
+ apidng — bojcvih — cnbkpe -f dmaflo — elmhjc — fkidng + gjaflo — ihbkpe 
— ihaflo + gjbkpe + fkcjhvi — elidng + mdbkpe — cnaflo — boidng + pahmcj 
+ apidfo — bojcep — nbkhm 4- cdmalng — elmnbk — fkalng + gjidfo — ihpecj 
— Hiding + gjkbmh + fkcjpe — elidfo + mdjcep — ncidfo — obalng +pahmbk. 
= dhonl — b % pgmk — dpfmj + ddonie — e 2 dpkj + f'Hlco + g 2 blni — h?amkj 
— ftpdfg + j' 2 oech + k 2 nbeh — Pmafg + ndblch — n 2 kadg — o 2 jadf + pHbec : 
we have, as usual, 
"u = (W - W) 3 Oi>2 2 - pdgdf <»•/ - ^V) 3 (pip? - pdpd) 3 ■ u- 
Particular forms of U are 
A = l, 5 = 0: 
u = ^ + 323 + 3(£> + 63B + 60 = (ap — bo — cn + dm — el + fk + gj — hi) 3 , = v suppose. 
A = 1, 5 = 9: 
u = a + 323 - 6<2D - 3B + 33<2B + 9§ - 18® - 27?% =9U suppose, 
where 6U = 0 is the result of the elimination of the variables from the equations 
d Xl 5=0, dyJJ = 0, d Zl 5=0, d Wl U = 0, d Xi 5 = 0, d y \J = 0, d Zi 5=0, d Wi U = 0. In fact, by 
an investigation similar to Mr Boole’s, applied to a function such as 5, it is shown 
that 6U has the characteristic property of the function u: also in the present case 
u is the most general function of its kind, so that 6U is obtained from 5 by 
properly determining the constant. This has been effected by comparing the value of 
u, in the symmetrical case, with the value of 05, in the same case, the expanded 
expression of which is given by Mr Boole in the Journal, vol. iv. p. 169. Assuming 
A = 1, the result was 5=9. [Incorrect: the result of the elimination is not 05 = 0, 
but an equation of a higher degree.] 
The general form of u now becomes 
u = a v 3 + /3d 5, 
in which a, ¡3, are indeterminate. 
We have u = av 3 + ¡3dU= M(av 3 +/3dU), 
where M = (W - W) 3 (pdg? — pjph 2 ) 3 (»iW - v^vj) 3 (p^pd - Pi l pp) s .