380 Proceedings of Royal Society of Edinburgh. [sess. 
1.4, 7 
2.5, 8 
3, 6, 9 
2,4, 7 
3, 5, 8 
1.6, 9 
5, 9, 10 
6, 7, 10 
4, 8, 10 
4, 5, 6, 7, 8, 9, 10. 
Consequently, if we wish to obtain a relation connecting certain of 
the variables, we select m + 1 of these equations involving the said 
variables and m others, so that we may be able to eliminate the m 
variables which are not desired and retain those which are. Thus, 
supposing we wish to obtain four equations involving 1, 2, 3, 10, as 
was the case in § 18, we take the 2nd, 3rd, 5th, 6th, 7th equations, 
which involve these variables, and four others, viz., 5, 6, 8, 9, and 
from these five equations eliminate the latter set of unknowns. 
One equation of the desired kind having thus been got, two others 
follow from it by cyclical substitution. The requisite fourth equa- 
tion, which transforms into itself by the cyclical substitution, is got 
by taking the tenth equation along with the first six, and eliminating 
the six variables 4, 5, 6, 7, 8, 9. 
20. The method applies not merely to the variables 
. . . ., which are of the 3rd degree. For supposing the equations 
in xy, yz^ zx (see § 6) are wanted, we have only to seek for an 
equation in xhj, xyz^ zx^, that is, in 1, 4, 10, 9, and, when it has 
been got, strike out the common factor x^ and derive the two other 
equations by cyclical substitution. The 1st, 6 th, and 8th equations 
enable us to do this, as they involve respectively 1,4,7; 1,6,9 ; 6,7,10; 
that is to say, only two variables (6, 7) besides those wanted. 
Thus, writing the three equations in the form 
’ » -1- kxy"^ ~ ^ I 
- 2B'ic%) + 'Pz‘^x = 0 1- 
- 2A'xyz -t- K.r?/2 + Bz^x = 0 J 
