[ =0 



1 62 Dr. J. BOTTOMLEY on 



If wc multiply this last equation by PX, and subtract from 

 (14), multiplied by Q — V, we obtain 

 -X«{Q(Q-V) + P(W-R)}+X2{R(Q-V) + P{Y + S)} 



+ X{S(Q-V)-P(Z + T)}-T(Q-V) = 

 eliminating X^ between this equation and (17), we obtain 

 the following quadratic equation for determining X : 

 X-^ 1 (R(Q - V) + P(Y + S))(Q - V) + (Q(Q - V) + P{W - R))^ 

 (W-R);-+x{(S(Q-V)-P(Z + T))(Q-V)-(Q(Q-V) 

 + P(W - R))(Y + S)} - T(Q - V)- + (Q(Q - V) 

 + P(W-R))(Z + T) 



If we write the solution of this equation in the form 



2A 2A ^ 



the following will be the values of the letters A, B, C 

 deduced from (16): 

 A = (fl2 - d"-y{x*{a"- - U'f + zx'iaY-ia' + U^) - {a" - U')\a' + ^^)) 



B = 2{a- - b'fd-x'i^ x\a" - b~f + x-(a:y-{2a" - b') -2(0^- b'fiir + r)) 



+ {ay- + {d'-b-'){d'-r)y} 

 C = {d' - b'fa'x-lx'id' - b') + .i--((2«-^ - b')f - 2{d' ~ b''){d' + r)) 



+ (y + a- - r){dy + {a- - b-){d' - r)) } 

 From the value of X thus obtained, we may deduce the 

 value of Y by writing in the formulae b, y, x for a,x,y 

 respectively ; these values of X and Y substituted in the 

 equation to the ellipse or in the equation 



{X-xf+{Y-yf^r 

 will give the equation to the instantaneous curve generated 

 by the dissolution of an elliptic cylinder. It will also give an 

 external curve cutting the normals at a distance c from the 

 ellipse. The radius of curvature at the extremity of the 



major axis of the ellipse has the value — , while c is less than 



this value, the internal curve cuts the normals drawn in any 

 quadrant in the same quadrant, when c is greater, the curve 

 becomes more complicated and assumes the form repre- 



