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

 and 



and determine the functions ^Ir. 



25. Denoting by c{<l>{i/}) the coefficient of <^{y} in tt, we see 

 that 



c{yt) = u^o^l, 



and all the terms in p* are symmetric, and 



-^.2/^=5./. iff) 



26. Let 7j denote one and 6 another of the quantities c)t,^,y, S, 

 and let 77, and 6^ be respectively equal to a"* and a". Then 



and the terms of the form yl y^ are symmetric, and 



27. Again: 



and the terms of the form yl yl are symmetric, and 



-^'Vi^yi^^-yx^yi W 



28. I shall next consider y\y<iy2,y\i of which the coefficient is 



S . «, {^2(73 + 74) + ^3(72 + 74) + /54(72 + 73) } • 



Developing this and making the substitutions indicated by (24), 

 we find 



<yiy22^32/4)=4 + 5(a + a2 + a-Ha^) = -l. 



29. If in the formulae of (28) we replace a by a^, a^, and ct^ 

 successively, we shall be conducted to respective results identical 

 with the last. In other words, 



AyiyiyzV^) =<^{y\y2,yby^-='<^{y\y4y<i.y-^'=^<^[y\yf;y\yz\ = - 1- 



30. In place of Y^ put a7^Y,.. The product will be unaltered 

 in value, but there will arise changes indicated by the symbolic 

 equation 



x^\y yi> ^3* y^y ys) =x(y6> ^u y^* ys^ yd- 



We hence infer that 



<y2y3y4y5)-f^{yiy2y3y4)= -^y 



and, all the terms of this form being symmetric, that 



^- 2/12/2^3^4= -2). yiy2y3y4 U) 



31. Lastly, let 



c{ylyry,)'=(9jr,s)^{q,s,r). 



