409] OE EQUAL ROOTS OF A BINARY QUARTIC OR QUINTIC. 311 
And the remaining seven Tables might of course be deduced from these by writing 
(f> e > G > b, a) instead of (a, 6, c, d, e, f), and making the corresponding alterations 
in the top line of each Table. 
18. The equations 31 = 0, 33 = 0,...., 911 = 0 consequently establish between the 
fifteen functions 1234, 1235, ...3456 a system of fourteen equations, viz. the first and 
last three of these are 
1234 = 0, 
1235 = 0, 
-160758675.1245 
+ 11559295.1236 = 0, 
+ 11559295.1456 
- 160758675.2356 = 0, 
2456 = 0, 
3456 = 0. 
To complete the proof that in virtue of the equations 2f = 0, 33 = 0,.., 9DÎ = 0 all 
the fifteen functions 1234, 1235, ...3456 vanish, it is necessary to make use of the 
identical relations subsisting between these quantities 1234, &c. ; thus we have 
a . 1345 + 46.1245 + 6c . 1235 + U. 1234 = 0, 
6 .1345 + 4c. 1245 + U. 1235 + 4e . 1234 = 0, 
which, in virtue of the above equations 1234 = 0 and 1235 = 0, become 
a . 1345 + 46.1245 = 0, 
6 . 1345 +4c. 1245 = 0, 
giving (unless indeed ac — 6 2 = 0) 1245 = 0, 1345 = 0; the equation 1245 = 0 then 
reduces the third of the above equations to 1236 = 0, and so on until it is shown 
that the fifteen quantities all vanish.