374 Proceedings of Eoyal Society of Ediribiirgh. [sess. 
tuting in (a), we obtain another set in the same variables. The 
result is 
B{2AQA'B'-C(ABC + PQE)} yz 
+ 2 (B2CC'2 - 2BQ A'B'C' + 
+ Q{2BBB'C'- A'(ABC + PQB)}ir 2 / = 0 . . . (^ 
and two others. 
10. Lastly, by eliminating xy from (/Ig) and (yg), we have the 
equations in it*, y, z, viz., 
2QE{(ABC + PQE)B' - 2CPA'C' }x 
+ 2AC{(ABC + PQE)A' - 2BEB'C' }y 
+ {4BCPEC'2-(ABC + PQE)2}2: = 0. . . (y). 
and two others. 
11. From these sets of equations several forms of the eliminant 
are obtainable. All of them are a little forbidding in appearance, 
and the calculation of the expansion of them may seem trouble- 
some ; but if attention be paid to the fact that any term leads to 
two others by means of cyclical substitution, the work will be found 
comparatively easy. ^ The same fact makes it also desirable to use 
a symbol, such as 2, for shortly expressing cyclical sums of three 
terms. Thus 
iBC . EP . C'2 
would stand for 
BC • EP . C'2 + CA . PQ . A'2 + AB • QE • B'2, 
where, it may be observed, the multiplication points are used to 
separate the letters of one cycle from those of another. 
12. Perhaps, for our present purpose, the best form of writing 
the eliminant, as thus calculated, is 
(ABC + PQE)^ 4- 64ABC • PQE . A'^B'^C'^ + 32ABC • PQE • A'B'C'(ABC 4- PQE) 
- 8(ABC 4 - PQE)(4A'B'C' 4 - ABC 4 - PQE)2(AB • QE • B'2) 
4 - 16(A2B2C2 4 - P2Q2E2)I:(A • Q • A'B') 
4-16l:(A2B2.Q2E2.B'4) 
- 8(ABC4-PQE)(ABC-PQE)2A'B'C'. 
