Mr. J. Cockle on the Method of Symmetric Products, 171 



or ^^ = ^A and fi^^a^P^', 



hence 



20. The equality of the coefficients of y<^ y^ and y^ y^ is ex- 

 pressed by 



S . u^Uc^a^^^^'X . ai/32/33^4=0, 

 or 



(«2+l)/Sl + K+l)^3 = 2.J=(«i + l)^r^ + (a3+l)/33""'- 



Multiplying the left of this equation by Va^u^, and the right 

 by ySi/Sg, and slightly changing the form of the result, we obtain 



{(l+0;S, + (l+a3-^)/33}^^3=(«l + l)/^3+K + i)A. 



Multiply the last equation into /Sg and /S, successively. The 

 respective results, combined with {d), give 



/3i2=a, and 3^ =01.3. 



21. Now, from 



we deduce 



(/9,^+/3n083'+A-VF-3=(/S. + ^r")083+/83-), 

 and hence arrive at 



(F-2)2 = F-3 + 2E + 4=r + 2E + 2. . . . (e) 



22. But we also have 



E2=(X./8)2=2;./Q2 + 2S.^,/e,=E + 2r; . . (/) 



and the elimination of F between (e) and (/) gives 



E4-2E3-9E24-2E + 8=0=(E2-l)(E2-2E-8), 



where the values of E are 1, —1, -- 2, 4. I here adopt — 1, and 

 from (/) find that 1 is the corresponding value of F. 



23. These values reduce (c) to the form 



and the values of s are the unreal roots of 



^^-1=0, 



of which, if a be one, the others are a^, a^ and a^. 



24. Let us now make 



Yi = 2/1 + a 2/2 + ^^2/3 + a^2/4 + As. 



Y^^zy^ + u^y^-i-u y^ + u'^y^ + u^y^ 

 Y^^^yi + u^-^-u^ + uy^ + u^, 



N2 



