Borchardt's Form of the Eliminant of Tioo Equations. 
451 
Further, towards the formulating of a recurrent law of formation for the 
new function we shall have the following identities : — 
^3 
a,) 
«3 
^4 
a, 
^5 
^5 
+ <^2^t3 I { ^4 + ^4 J I + ^2<^a\ I ^3 + 
+ a^a^ 1 1 ^3 + ^4 [ I + <^2<^3«4, 
(a, + ^3+^4+^5)1153 64 h 
c 
d 
+ 2^,^31154+^4 ^s+C, 
d. 
+ ^a^a^a^ I { ^5 + + ^5 } I + a^a^a^a^, 
and so forth. With the double-suffix notation for the elements, or with 
the umbral notation, the details of the law are still more readily grasped. 
Thus the next identity of the series may be written 
I3 1. 
le\ =13 + 13 + 
56 
. + l6) I 1232423 
84 85 85 I 
4545 
+ SU3I {24 + 84 23 + 85 25 + 85^ 
+ 21,13141125 + 85 + 45 26 + 86 + 461 
56 J 
46 
56 
+ 21,131415(26+86+46+56) 
+ 1313141516, 
where we observe that by deleting the first element of the co-factor of 1,, 
and then combining the 1st and 2nd frame-lines by addition we obtain the 
co-factor of I2I3 ; and by treating similarly the co-factor of I2I3 we obtain 
the co-factor of I2I3I4 and so on. 
7. The development being arranged according to products of elements 
of the first line, what is required for practical purposes is a rule for telling 
the co-factor of such a product ; and this is easily formulated. Thus, 
if the co-factor of I5I6 be wanted, we note that the indices not found in 
I5I6 are 2, 8, 4, and thence form the triangular array 
23 24 
84, 
thereafter prefixing to this array the line of three elements got from taking 
2 along with 5 and 6, 8 along with 5 and 6, and 4 along with 5 and 6, the 
result being 
1(25 + 26 85 + 35 45 + 4,) 
23 2, \ 
3. ' 
