niet J 
XXV. Investigation of Brianchon’s Theorem. By STEPHEN 
Fenwick, Esq., Royal Military Academy, Woolwich*. 
iB a hexagon be circumscribed to a conic section, the three 
diagonals which join the three pairs of opposite summits 
will pass through the same point. 
This theorem, I believe, has not yet been completely esta- 
blished by the method of coordinates. Sir John Lubbock’s 
investigation, which appeared in the Philosophical Magazine 
for August 1888, is adapted to the case of the parabola; and 
the mode of extending it to the other cases is pointed out. 
In the following investigation, the equation of the conic sec- 
tion in general is used, and the property is demonstrated on 
elementary principles alone. 
Let 75 Po» Pg &c. be the points of contact, which we shall 
denote by 
U1 Yiy Lo Yo X3Yg9 V4 Yo V5 Ys Xe Yos 
and A B, BC, C D, &c. the tan- 
gents at these points. Having 
joined C F, EB, intersecting in 
O, and also. D O, AO; refer the 
system to C F, B E, as axes of co- 
ordinates, and denote the conic 
section by the general equation, 
ay?+bayt+ceu?+dyt+ex 
+f= OS tee ote a eee A. 
The several] tangents will then be 
denoted by the equations, 
(AB)...y (2amtba+d)+2 (2ex,4+by,+e)+dyteat2f= 0...(1.) 
(BC)...y (2a yotbx,+d) +2 (2 cr,t+bytetdytert2f= 0...(2.) 
(CD)..:.y(2ayg+b03+d)+a (2ex3+byzt+e) +dyzte r3t+2f = 0...(3.) 
(DE)...y (2aystba,+d)+a (2eaytbhy,te+dyste rgt+2f= 0...(4) 
(EF)...y (2ay;+b2,+d) +2 (2eastbu,te)+dy,ters+2f= 0...(5.) 
(F A)..-y (2ayo+6 ag td)+x (2c 16+6 yote) +dygtergt+2f= 0...(6.) 
Combining (3, 4.), and also (1. 6.), so that the absolute terms 
in the resulting equations may be eliminated, we get the equa- 
tions of D O, AO. 
(D 0)... 2aystbaztd  2ayy+bay+d\ f 2ea+byste Feet EY o (03) 
dyster,+2f  dy,pers+2hJ * dystemt2f dyipery42f 
(A 0). { Feutents Payot antl ee courer the . ieee = 0.4 (8) 
dyjteat2f dyetexrgt2f Ldytemt2f dystea+2f 
It will suffice now to show that the products of the coeffi- 
cients of y, « in (7.) and (8.), taken in a cross order, are equal 
* Communicated by T. 8. Davies, Esq.; Royal Military Academy. 
