124 Mr. J. Cockle on the Resolution of 



filters through the rocks from the Atlantic Ocean in the levels 

 which are worked out under the bed of the sea, is found to have 

 lost much of its original saltness. These facts are at your ser- 

 vice. I trust you will continue the interesting inquiry which 

 you have commenced. 



" I am, dear Sir, 



" Yours most truly, 

 " H. M. Witt, Esqr " Robert Hunt/' 



XVI. On the Resolution of Algebraic Equations of the Fifth 

 Degree, By James Cockle, M.A,y F.R.A.S., F.C.P.S., 

 Bannster-at'LaWj of the Middle Temple^, 



81. npHE last paper of ray series on the Method of Sym- 

 -i- metric Products terminated at p. 138 of this Journal 

 for Februaiy 1854 (S. 4. vol. vii.). The numbering of the para- 

 graphs is resumed here. 



82. The problem of (79) may be solved thus. Let Y^ vanish, 

 and let the quintic in y be 



S^-5B2/3_5C2,2_5j)y_j;^0. . . . (m) 

 Determine % and 11 from the quadratics 



Be2+(e-C)a=o, 



C(e2 + B3)-(D + B2)(B0-|-Il)=O, 

 and Yj, Y3 and Y4 from the equations 



Y3Y,=B, Y,Y3^=0, Y3^=Xi, 

 then the expression 



a Yg + «% + «% 

 will give all the roots of (/w). 



83. If, for the development of the right-hand side of the equa- 

 tion of (66), we employ the notation of (25) and the '^ epimetric'' 

 symbol of (69), and, further, if we make 



c(QrQ,») =c'(QrQ2") -5c''(Q,-Q/), 

 where c' is a symmetric and c" an unsymmetric function, we 

 have 



c"(Q,4) = -5-'6(^) = S,(^i2^A). 

 c"(Qi3Q,) = 2Si(^,8^2^3)-S,(^,V^32), 



(/'(Qi^Q,^) =2S,(^i V«^3) + 81(^,^2^3), 

 ^"(QiQa') = JiSi(«^i%V) + Si(^iV«^3)> 



* Communicated by the Author. Mr. Cockle adds, that his " expres- 

 sion for Mj (see his ' Fragment/ &c., Phil. Mag. for December 1853,p.448, 

 par. XX.) must be amended by the addition of the quantity — x("~y6) to it." 



