On the Equation of the Fifth Order. 211 
Vet P\LatPo@atps= t+ uy, 
apt peat pote t+ps= b+eu, 
@y + Pt, + poly + py=Ct+ Pu, 
25 + py X5 + po%s+pz= 3¢+ vu, 
Het p Let pote tpg t+ w. 
If from these equations we seek for the values of p,, Po Ps t, Uy 
we have 
By: Jo? 75:—t:—u=ll,: 4:1: T,: Wy: ly 
where IJ,, II,,.. denote the determinants formed out of the 
matrix 
cc mec ame Aiea ee 
v Bp x ee LBs Des Dy 
ae a Bo hr akgco ees 3 
& 3 ? U5, cS Wea Re uP 
& 2 Cpe Bagi bcd Gea 
2.e., denoting the columns of this matrix by 1, 2, 3, 4, 5, 6, we 
have 11,=23456, T,= —34561, H,=45612, &e. In par- 
ticular, the value of II, is 
— eer @udae bs 2, Loge 
aT tine OE a 
PISA NG 
x 33 a; 1, e 2 
Ue Ras ie AA 
And developing, and putting for shortness {a8} =x,49(%.—4p); 
&c., we have 
Tl, =( {a8} + {By} + fyot + {det + feat )\(—26+-—8 +204) 
+ (faryh + {yet + feBt + 180} + {Sat )(+e4+22—23—204), 
And this is also the form of the other determinants, the only 
difference being as to the meaning of the symbol {a6}, which, 
however, in each case denotes a function suchthat {apt =— ; Ba} : 
Writing for greater shortness, 
{oBryde} = faB} + {Bry} + {y8} + {oe} + feat, 
and in like manner : 
ee dens ca taeies Wict mune is mat lane Bas cs 
I], is an unsymmetric linear function (without constant term) of 
| P2 
