468 Proceedings of Royal Society of Edinburgh. [july 18, 
the proof of which is given as follows. In all the terms of f 1 m, 
every one of the integers except one occurs as the first integer of a 
couple, and every one of the integers except m occurs as the second 
integer of a couple : consequently, in every term of \n . f lm, the 
first places of the couples are occupied by the integers from 1 to r 
inclusive, while in the second places, m is still the only integer 
awanting, and n occurs twice. Suppose then all the terms of 
lmflm + 2w.f2m + ....+ rn . frm 
so written, that the first integers of the couples are in ascending 
order of magnitude, and let us attend to a single term 
..... -pm ...... • qn • 
in which the two couples, having n for second integer, are the 
y» th and q th . If we inquire from which of the expressions 
In. f lm, 2mf2m, .... this term comes, we see that it is a 
term of both pn. fpm and qn. fqm, and must, therefore, occur 
twice. Further, we see that mpn.fqm it has the sign of the term 
• pm • • qn • 
of the common denominator, and that in qn . fpm, it has the sign of 
the term 
..... 'pn> ...... • qm • ...... 
of the common denominator. But these two terms of the common 
denominator have different signs : consequently 
ltt.fl m + 2mf2m + . . . . + rn.frm 
consists of pairs of equal terms with unlike signs, and thus vanishes 
identically. (xn. 4.) 
These preparations having been attended to, the set of r equations 
with r unknowns is solved by Laplace’s method ; and a verification 
made after the manner of Vandermonde. It is also pointed out, 
that if the solution of a set of equations, say the four 
ax A + bx 2 + cx 3 + dx A = s 1 ] 
ex 1 + fx 2 + gx 3 + hx A = s 2 j 
ix x + Jix. 2 + lx 3 + mx A = s 3 | 
nx 1 + ox 2 +px 3 + qx A =s A J 
