Mr. J. Cockle on the Method of Symmetric Products. 495 



ducts to the higher prime equations will find a type in its applica- 

 tion to those of the fifth degree ; and the questions incidentally 

 suggested in the latter case will equally arise, perhaps in a more 

 general form, in the discussion of equations of the higher degrees, 

 WTien the degree is composite, simplifications of the processes 

 will probably be obtainable. But I shall here confine myself to 

 quintic equations, in the theory of which the following questions 

 now present themselves : (1) Is there a symmetric product ? and 

 (2), if not, does our search after one suggest an unsymmetric func- 

 tion wth any peculiar properties? 



8. Retaining the assumption m = ra— 1, and continuing to 

 replace v by y, let 



■^4(2/5) = C4C4 + ^4 = 1*4 + U4, 



where P4 is symmetric and U^^ evanescent or unsymmetric, and 

 it remains to be seen whether R4 is equal to U4. 



9. It is first to be remarked, that f{x) may always be deter- 

 mined so as to reduce the equation in y to the form 



f+9sy^+q6=o (b) 



And when this relation subsists P4 becomes equal to zero. 

 Hence, if a symmetric product exist, there will be no difficulty 

 in making it vanish. If not, we may always assume that 



7^4(2/5) = U4- 



For effecting the transformation {b) we may avail ourselves of 

 Mr. Jerrard's process, or of the more convenient one which I 

 have given for the purpose (Phil. Mag. S. 3. vol. xxxii. pp. 50,51), 

 and in which the solution of 0(a;, ?/, 5') = is supposed to be 

 effected by the Method of Vanishing Groups. 



10. The quantity P4 may be expressed by 



X.y' + ^^.y,^y, + YX.y,'y,^ + GX.y,hj,y, + }i^.y,y,rj,y,; 



and, guided by the analogy afforded by the application of the 

 method to biquadratics, I shall first proceed to inquire whether 

 •Tr/y^) can be made to take the above form. If not, our object 

 must be to reduce the unsymmetric part U4 within the narrowest 

 possible limits. Whether the process which follows be the most 

 advantageous for our purpose may be a subject of future and 

 formal inquiry. But its strong prima, facie claims warrant its 

 adoption here. 



11. As well to fix our ideas as to facilitate our operations, let 

 us write 



Y,=yi+«,y2+ ^1^3 +7)2/4 +^1^5. 



Y2 = y, + «2y2 + ^2^3 + 72^4 + ^2^5. 

 Y3 = yi + "3^2 + ^3^3 + J3V4 + hVs, 

 ^4 = ^1 + «4y2 + ^4^3 + 74^4 + ^4^5* 



'r4(y6) = Y,Y,Y3Y4. 



