Mr. J. Cockle on the Theori^ of Equationi. 293 



jno-il biDXi onorf^jsoflq sjit sJfiJiqiaojq oJ bs'iiopai arnil to vjii 



tHS'yUffek^is of J from 9, 8, 7, 6, into 2, ^j 6, 8, respectively, 

 for we may groir}i as we please the squares into wliich-^?i(9j:) 

 is decomposed. This gives us after reduction J^'^^ == vui;) 



.tcffj hyni/Q 1=72 -2 + 78^a5biTiy -iij^fe^ +70* b ^arfig •.8.9^*) 

 -rvJo ^iJ«9'f§ ad ^^fiffr^obcfn naa^fiado nfid? -v-offr -.ofi 93119^3191 



(if 



the terms which (after substituting for ^3 in/^(9.) its value 

 derived from (16.)) contain 2,/, then, a^ being the coefficient 



^(.§fi? .k,t^ rm\k. ., ,>9i..i,i .-n,9d 



b9x^i/;d9 ^<,l§n8-ij8 eamoyjjJ j ? \ y.J Vya^ H ft annu bi-iiuq 



'; '*nfiy*ve'''^'gefeM,^matever be the VBXWm^^m^^ 

 j/^(10.), and is applicable to the annihilation of the 2ndj 4th 

 and 6th terms of the equation of the 6th degree; but, in case 

 the order in which the squares into which /^(9.) is decom- 

 posed are to be grouped should in any instance become mate- 

 rial, the last paragraph will have to be modified accordingly.- 

 '"''''7: [Supplement to p. 384 of the last volume.) — Let n = 4, 

 and add a term, N<p (.?■'), to the right-hand side of (3.) ; then 

 a product similar to (4.) may be obtained by means of the prin- 

 ciple, that "(Pi, <P25 " • <pn-i being linear and homogeneous 

 functions of Aj, Xc^ x^ and (p^*"^ denoting the result of substi- 

 tuting .v'-''^ for A' in f ; then, when tt is composed of symmetric 

 functions o^ x, &c., tt'' (=;S{<Pi<P2' • 'P'»-2<P"«-i}) consists of 

 symmetric functions of .Ti . . .:p'i . . . x''^ •* " so that, when x"=x'f 

 ir^^ = 9r". By means of the obvious extensions of this principle, 

 v/hich is a branch of one of numerous classes of the same kind 

 as that used in paragraph 3, we may add to (3.) as many terms 

 as we please, for any values of Ji. These remarks are made 

 with reference to paragraphs 5 and 6. It is possible that 

 in all these investigations we may be able to derive some aid 

 from the quaternion theory, a different distribution of a, &c. ; 

 and it willbe borne in mind that a want of symmetry in the 



