NOTES AND REFERENCES. 
611 
The two equations are 
ciwt 5 + 5e(w — 100 ae) r + wf= 0, r (w - 20 ae) + 20 a/= 0, 
or, if for greater convenience we write 20 ae = 0, then 
aiVT 5 + 5e (w - 0) r + wf= 0, r (w - 6) + 20 af = 0, 
we have 
20 af 
T ~~w^~d’ 
and thence 
— 3200000 a*/ 5 100 aefOw — 50) . „ 
aw - („-tf uT-e +w f=°- 
that is 
w {w - dy -50 (w- 50) (w - Of - 3200000 a 6 / 4 w = 0, 
which should be identical with the before mentioned equation in w, that is with 
(w s - 50w- +150-w + 5 0 3 f - (3200000 + 256 0 5 ) w = 0, 
and it is in fact at once seen that each of these equations is 
0 = w G 
+ w\- 10 0 
+ w 4 . 55 0- 
+ w 3 . — 140 0 3 
+ w-. 175 0 4 
+ w . -106 0* - 3200000 a 6 / 4 
+ w\ 25 0 6 ; 
which completes the proof. The proof for the general form (a, b, c, d, e, f\x, y) G is 
similar in principle, viz. treating for the moment <£ 2 or w as a constant, we have 
in t a quintic equation and a cubic equation, in each of which the coefficients con 
tain w linearly; and the elimination of r leads to the required sextic equation in w, 
but there would probably be considerable difficulty in effecting the calculations. 
It thus appears that assuming the solution of the central resolvent equation 
for t, = ^, we also know </> : I recall that for the quintic equation whose roots are 
x u x 3 , x it x 5 , the .significations of these quantities are 
where 
X_ (12345) -(24135) 
I " 12345 - 24135 ’ 
4> = 12345 - 24135, 
12345 = 12+ 23+ 34+ 45 + 51, meaning thereby x x x 3 + x.x 3 + x 3 x A + x 4 x s +x 5 x 1 , 
(12345) = 123 + 234 + 345 + 451 + 512 „ „ XjX 2 x 3 + + x 3 pc i x 5 + x^c 3 x x + x 3 x y x 3 , 
77—2