54 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Lemonnier, H. (1875, 1878). 
[Theoremes concernant les equations qui ont des racines communes. 
Comptes-Rendus .... Acad, des Sci. (Paris), lxxx. pp. 111-112, 
252-255.] 
[Memoire sur 1’elimination. Annates de Vlfcole Norm. Sup. (2), vii. 
pp. 77-96, 151-214.] 
Lemonnier’s condition for the equations 
%x m + ■ • • + a m = 0, b 0 x n + ••• + &„ = 0 
having k common roots is different from Trudi’s, but fortunately for com- 
parison is very easily expressed in Trudi’s notation. It is * that the first 
k determinants of 
: K> ' ' 
• 5 ®m)n—k + 1 
1 (&<)>•• 
shall vanish, and the first determinant of 
(“o. • • 
• ? ®>m)n—k 
• j l*n)m—k 
shall not vanish. The former part of the condition recalls Zeipel’s of 1859 : 
the latter is an important necessary adjunct. When, however, the equation 
of the common roots 
\(x k , a* -1 , . . . , a? 0 ) = 0 
{ a Q > • • 
• 5 a m)n-k j 
• j bfim-k 1 
happens to be given along with the condition, it is less necessary to mention 
the latter part, as the determinant involved is the coefficient of x k in the 
said equation. 
Muir, T. (1876). 
[New general formulae for the transformation of infinite series into 
continued fractions. Trails. Roy. Soc. Edin., xxvii. pp. 467-471.] 
[On the transformation of Gauss’ hypergeometric series into a continued 
fraction. Proc. London Math. Soc., vii. 112-118.] 
The fundamental theorem, which is established in two different ways, 
is not essentially different from Heilermann’s of 1845 (Hist., ii. p. 361). 
The second of the two ways is the more interesting. Beginning with the 
series 
a 0 + ape + ap? + ap? + ••••, or / 0 , 
b 0 + bp + bp? + bp 3 + ••••, or fi , 
* This is in accordance with the statement in § 13 of the complete memoir, and is 
somewhat different from that first published. 
