( 105 ) 
At the 8th Physical and Medical Congress held at Rotterdam in 
the month of April last I gave a communication in the sub- 
section for pure and applied mathematics concerning the relations 
existing between the roots of # homogeneous equations with n + 1 
variables and the coefficients of these equations, and I have pointed 
out ia. that for such a system of equaticns the following property 
exists: the products of the corresponding elements of the » systems 
of roots, satisfymg this equation, stand in the same proportion as 
the resultants furnished by the system of equation, when every time 
one of the variables is made to disappear. 
This property provides us with the means of expressing the 
coordinates of the ninth point in the coordinates of the 8 given 
points. To do so we apply it to the system of equations (1) as follows: 
Li Tg e ° dg Lg Yi Vg 5 5 Ys Yo ry Sn £g . : 28 <9 
/ Üz=0 R y=0 LR 20 
where 9, 49, 29 represent the coordinates of the ninth point and 
Ro, Ryo, Rao the indicated resultants. 
We now obtain when paying attention to the signs + and — by 
which the resultants are affected: 
Tj Lg Ce ie dg Tg —— KAI Vg eene Us Y9 21 23 a CN 23 29 
= == (6). 
ay by ty) bi aj bi 
ag bg ay by Me ba dj bi ag bo aj b, 
dg by ag bs dy by 16 be as bs ay by a4 by ao by a) b, 
Ajo Dio ag by ag bg a9 b 10 are be as bs Un by Ay by ag by 
110 Dio dy by dio bio a6 be ay by ay, bs 
J 
dio Oro aio bio a, by 
* 
The denominators of these fractions can be developed according 
to products of determinants contained in the assemblant (2) which 
may be replaced in consequence of the equation (4) by determi- 
nants derived from the assemblant (3). 
