Joi.Y — Vector Expressions /or Cur res. 387 



is the equation of the projected curve, and as the constant function t/^ 



and the operator 



d d 



dxx dyx 



are commutative in order of operation, the emanant curves project into 

 t^manants of the projected curve. 



The emanants of any order p defined by x^yx, have the same tangent 

 line and osculating plane at the point 



at which they meet the original curve. For, at any point x = Xi^ V^y-i 

 on the emanant, the tangent line is 



Id d\l d dY\r ^ 



P = 



id d\( d d y-' .. ■ 



and this becomes identical with the tangent line at Xiyi to the original 

 curve when x^^Xi and y% = yi. 



In like manner, the osculating plane at X2y-i on the emanant is 



d^ d? d^\( d d y-^ , , , 



^ ' d' d~ d^\i d d y-' ^ , ' 



and this is, when x^ = x^, and yi = ya, the same as the osculating plane 

 at Xiyi to the original curve. 



13. General properties of the emanant curves. 



The emanant at x^y^ of order p intersects the emanant at x^y., of 

 order n-p. In fact, 



is a point common to the two curves. Again, as the equation of the 



