806 
m. H. J. S. SMITH ON SYSTEMS OE LINEAE 
will represent a set of fundamental solutions of (43.). For, in the first place, it repre- 
sents a set of independent solutions ; because its first determinant is x ^ X • • . 
or T^, or — ; therefore its determinants are proportional to C„ Cj, . . . &c. ; or since 
A,,+i c 
Q 
the first of them is -j, they are respectively equal to the numbers 
C, C2 C3 
^5 • • 
c c c 
which admit of no common divisor. 
To obtain a set of values for the constituents which occur in the matrix (45.), we 
may form the series of equations 
^1 — Aj 
^2 C3 + ^2A2=A3 ^ 
— 1 1 f^»An i _ 
(46.) 
It will then be found that the equations (44.) are satisfied by the values of r comprised 
in the formula ^ 
- V. ; (47.) 
and on substituting these values in the matrix (45.), it will coincide, after an unim- 
portant modification, with that occurring in M. Heemite’s solution of the problem. 
But, in practice, the simplest method of obtaining a solution of the problem con- 
sidered in this article, is to solve the equation (43.) by Eulee’s method, and to employ 
in the place of the matrix (45.), the matrix of the set of fundamental solutions thus 
obtained (see art. 6).^ 
Art. 10. Another problem, closely connected with the preceding, and of no less fre- 
quent application, has also been completely solved by M. Heemite * ; but as it may serve 
to illustrate the utility of the methods employed in this paper, we shall venture to resume 
and generalize it here. The problem is 
Given a set of n-\-\ numbers Cj, C 2 , . . . C„+, without any common divisor, to assign 
all the matrices ||a:|| of the type n'x{n-\-l) which satisfy the equation 
X 
Let c„ c^. . . . be any particular solution of the equation 
+ • • • • -l"C»+iyn+i = l (48.) 
(which is always possible because C,, Cg, . . C„+j are relatively prime); and let ||y|| repre- 
sent a set of fundamental solutions of the equation 
<?.3'i+G2/2+ • • • (49.) 
* Liotjtille, vol. xiv. p. 21. 
