396 



such a change of origin as would make the axis pass through the 

 other extremities of the lines. 



8. Now let X O 2 . . . O n O 1 be a completely enclosed polygon, the 

 sides of which, when produced, cut the radical loci of a known alge- 

 braical equation of n dimensions in known points. Let the product 

 of the series of ratios formed as above (3), corresponding to lines 

 drawn from the point O r in the direction O r O r +i (on which last- 

 named line the unit line Q r O f r+i> corresponding to OI in (3), is mea- 

 sured), be represented by 



[O r , r+1 M-rO r O' r+1 ]. 



And let X r>r +i be the coefficient necessary to make this product a 

 correct representation of the formation on the left-hand side of the 

 given equation for the values of x and y, corresponding to the point 

 O r and the direction O r O r +i. 



Then for the point O r +i and the direction O r+ iO r (for which the 

 corresponding unit line is O r +iO' r ), we shall have the product of the 

 corresponding series of ratios represented by 



[Or+l, r M-rO r +i O' r ], 



where 



Ar+l,r=( l) W ^r, r+1- 



Proceeding then round the polygon, and forming the products for 

 each of the two sides terminating at each point, we have by (7), 



Xi, w [0 1)W M-f-OiO' w ]=X 1)2 [0 1)2 M^0 1 0' 2 ] 



(-O nx i,2[0 2 ,iM~0 2 OY) = X 2j 3 [0 2 , 3 M-H0 2 0' 3 ], 



[O n ,n~i M-rO w O'J =X n>1 [0 fl M-rO.O',]. 



Hence multiplying all these n equations together, and remembering 

 thatX 1>n =( 1) W .X M)1 , and that ( !)" = ( l) w , since nn and n 

 are both odd or both even, we have 



=(-!). [0 1 , 2 M^0 1 0' 2 ].[0 2)3 M4-0 2 0' 3 ]..'.[0 W)1 M-f-O n O' 1 ]. 



Now since O rjr+1 M and O r +i, r M are on the same straight line, 

 their ratio is scalar j and since O 1 O' /l = 1 . O W O',, we have, by an 



