194 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
U = 21 ay (e223 +23 Ls) = 23 U4 05=5*e ; 
. by (B’), 0,=-5U,=—5’e; 
and similarly 
,=6,=—5 6) G0; == 5, end 0;=-— 5; 
. 00503 0,0;,0,=5*. 
But in the particular case now under notice 
B=0;andp=Q'= 535: '3..0).050, 00:0, =k = BEES 
consequently Sk = br ero 
Generally, therefore, 
0, O, 0; 0, 0; 0, = 5" (PQSEH? + «E?+2AE+ p), 
which gives for (d), by restoring the values of «x, A and pw 
and making P and § vanish, 
0, 0, 0, 6, 6; 0g= 5% Q’. 
Section III. 
Direct Calculation of the Equation in @. 
28. The six values of @ may be regarded as the roots of 
the equation 
+D,+D,64+D;4+D,0+D;0+D,=0 . (1). 
Suppose now that, by Mr. Jerrard’s method, or otherwise, 
the general equation of the fifth degree is reduced to the 
form 
v—-5Q2°+E=0 . . sepa 
then (Art. 26) 
6=-5r=-—5U, and 
(Di = 20527 =—b>, Us 
We might, therefore, calculate this coefficient by writing 
out the six values of —0@, 57°, or 5U, and taking their 
sum; but the labour may be materially abridged by the 
following method. (See Section II., Art. 18, et seq.) 
TS Fa) — > ay wes, 
= 2 (TLR L, Wy Ly + Ly Ny Ly Ly +H, Ly Lys + Aj Ly Us) 
= Sy" (a? 23+ Qa} a, L;) +2DH, Wy Ly Xq3 
or, since 22, #2 2, 2,=0, 
T= Dy (a H3 +277 a 2). 
