OF LOCAL THEOREMS AND POKISMS. 



343 



Any curve of this kind being given, if we take two equal ah- 

 seisscB, on opposite sides of the centre, and prolong the ordinates 

 at both these points, until the parts of each produced be equal to 

 the ordinate at the other point, then the locus of the points thus 

 found is a right line given by position. 



If we make <px = a-\-x, the curve becomes a right line, 



c 



whose equation is i/ = - + - a; ; and the line found is paral- 



lei to the axis, and the curve given, is a right line passing 

 through a point in the axis of the ordinates, equally distant 

 from each parallel. 



Let ABC be any periodic curve. Fig. 5. and DB, EC, any 

 two corresponding ordinates ; draw the ordinates FD, GC, 

 and draw the line HFG, passing through the points F and 

 G, it is required to find the equation of this line. 



w and V being the co-ordinates of any point in this line, its 

 equation will be v — Aw-\-B; the quantities A and B being 

 determined by the condition, that it must pass through the 

 points F and G, whose co-ordinates are respectively x and | x, 

 and a, X, and -^ ux, we have 



4" a; — 4* a * 1 « * . 4' ii^ — x.^ux 



V = 



■w -)- 



— uX X — OCX 



Let us now determine the form of the curve AFG by this 

 condition, that the line HG shall always be parallel to itself. 

 It is well known, that the co-efficient of the abscissa w ex- 

 presses the tangent of the angle of the inclination of the line 

 to its axis ; this must therefore be constant, and we have 



^x ^a_x __ ^^ ^^ ^^ — 'l'ux=.c(x — ux)i 

 X — aX 



the solution of this equation is ^ a; = \^j^^;^^ . 



This suggests the following porisms : 



VOL. IX. p. ir. X X Any 



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