174: Mr. J. Cockle on the Method of Symmetric Product^, 



and making 



E=-l, F«l, G;=2, H==-l, 



the relation given in (10) becomes 



P4=S . /-S . yi^ya+S • yiW+^^ • y^y^Vs-^ • yiy^y^yi- 



39. And we also have 



v^^-6^'.yi\y^ys,-hysy4}=-^{yi%y2yb+yBy4) 

 +y<i{y\yz'^yAys) +yB^iyb+y2yAi+yA{y\y<i+y^b) 

 •^y^^yiyA-^y^y^' 



40. If we denote by S a symmetric, and by U an unsymmetric 

 function, the equation 



is satisfied, not only by 



S=P4, u=u„ 



but also by 



S=P4-5S . y^y^y^y U=U4+ 52; . y^^y^y^ 

 and in other ways not necessary to be mentioned. 



41. In effecting the transformation {b) of (9) by my process, 

 the second condition 



must be superseded by the equation 



b,-|b,B3+^b,^b,-^b.^=o. 



the left of which is a critical function. Each of the functions Y 

 and, consequently, their product is critical. 



42. It does not appear to be possible to render ir^f^y^ perfectly 

 symmetric in the general equation. For, starting with the 

 results of (20), we may convert 



c{y^yiys)-<yxy2ys^) = {^^ i. 3)-(3, i, 2)=o 



into 



<t>'(\)<t>'i^/fi) + cl>'( \^\)ii>\ti) - 2(t>'{ ^X)<^'( x/fi) =0, 

 where 



<j)'{r)=r-^r~^. 



But if we combine this relation with 



c(y.' y,) = cCyiVs) 'or f W +<t>'W= 4>'{ ^^) + <#>'( V/t), 



we shall be led to conditions which seem inconsistent with sym- 

 metry. 



43. In line 8 of (4) Pn must be replaced by Pn-i, and in line 

 14 of (5) C„_i must be put in place of C„. 



2 Pump Court, Temple, 

 Pec. 24, 1862. 



