588 ON THE CONDITIONS OF EQUAL FOOTS OF A BINARY QUARTIC OR QUINTIC. 
And the remaining seven Tables might of coarse be deduced from these by writing 
(/', e, d, c, b, a) instead of (a, b, c, d, e, f), and making the corresponding alterations in 
the top line of each Table. 
18. The equations 91=0, 15=0, . . . ., J$l=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 $=0, 33 = 0, . . , i¥l=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 
« . 1345 + 4 b . 1245 + 6c . 1235+4^ . 1234=0, 
b . 1345 + 4c. 1245+6<L 1235 + 4c . 1234=0, 
which, in virtue of the above equations 1234=0 and 1235=0, become 
a. 1345 + 45. 1245=0, 
b . 1345+4c . 1245=0, 
giving (unless indeed ac—b 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. 
