475 
sketch, and on which he hopes to return in a future commu- 
nication. 
- Theorem.—Let A,, Ay ..- An be any » points (in num- 
ber odd or even) assumed at pleasure on the n successive sides 
of a closed polygon BB,B, ..- Bn-1 (plane or gauche), in- 
scribed in any given surface of the second order. Take any 
three points, P, Q, 2, on that surface, as initial points, and 
draw from each a system of n successive chords, passing in 
order through the n assumed points (4), and terminating in 
three other superficial and final points, P’, Q, Re’. Then there 
will be (in general) another inscribed and closed polygon, 
CC,C, ..+ Cn-1 of which the sides shall pass successively, 
in the same order, through the same n points (4): and of 
which the initial point C shall also be connected with the 
point B of the former polygon, by the relations 
acl By _ del By, 
be ae ‘co aeN’ 
afin ya _ Uf oe 
oa Ble ea BEM 
ab ynv ab’ ynv 
. where a= QR, b= RP, c=PQ, 
: \ e¢= BP; f= BQ g = BR, 
l= CP, m= CQ, n=CR, 
a= QR, v=RP, c= PQ, 
e'= BP, J’ = BQ, gj = BR, 
ACP, m = CQ, n' = CR’; 
while aByeZnAuv, and a‘P'yeSXuv’, denote the semidiame- 
ters of the surface, respectively parallel to the chords abcefglmn, 
ace f'gimn. 
As avery particular case of this theorem, we may sup- 
pose that PYRP’QR' is a plane hexagon in a conic, and BC 
its Pascal’s line. 
