484 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



From these formulse it appears, that we may interchange x^ and x, provided 

 that we at the same time interchange y^ and y, and that we may also interchange 

 x,_ and x„ and at the same time y^ and ?/, ; so that, in fact, they are reducible to 

 two, vi^. 



^^y^-^yVz =x,y-^y^' C'') 



x^x^ + cy^y^-x^x + cyy„; . . . . (x) 



and here x, y ; x, y, denote generally the co-ordinates of the extremities of either 

 of the chords, and x^, y, ; x., y. those of the extremities of the other chord. 



These two equations contain the four pair of co-ordinates x, y ; x^ y^, &c., and 

 only the simple power of each ordinate. Therefore, each may be deteraiined by 

 the common process of elimination. And, to give the formulae the greatest degree 

 of simpMcity, we must recollect that 



a' + cy = a', i^^_ + cy\ = a\ 



«, + cf^ -a?, x\ + cyl = a\ 



The results obtained are 



d'x = cy^ (x^y^ + x^j^) + x^ {x^x^ - cy^^) \ ^ 



a'y - «, {x^^ + xj/,) -y, (^,«2-ey,i'J f ' 



a'x= cy (x^y„ + x^,) + x (x,x-cyy^ \ ^ 



a^y= X {xy„-^x.^:) -y (x^x^_- cy^y.^ J ' 



«'^, = <:y^ (»^y, + ^^ ) + ^2 (^«3 - '^yy^ \ . ^ 



«'.?', = ^, {.xy, + x^ ) ~y, ixx^ - eyy^) j ' 



«X = cy, (xy^ +xjf)-{-x^ (xx^ - cyyj "I 

 a'y, = z^{zy, +z^y ) -y^ {xx^ - cyy^ j ' 



These formulae completely resolve the problem that was proposed. 



Fig. 4. 

 Fig. 3, 



E p 



4. If one of the chords PP'" be supposed to pass through A. one extremity of 

 the axis CA, so that x^^a and 2/3=0, then we have 



