JoLY — Vector Expressions for Curves, 375 



are of the order n. The point on the curve determined hy x = Xi and 

 y = yi may be called the point Ziyi. 



Vector expressions for the tangent line and the osculating plane at 

 Xi^i may he readily assigned. 



The equation of the tangent line is 



d d 



X — - -r- y 



dx-^ difi 



^(x^i) 



""k^'-^iiy''"'^'^ 



where x^yi are given, and xy variable. This is in fact the equation of 

 a right line passing through Xiyi, as appears on putting x = Xi and 

 y - yi, and also passing through the consecutive points Xi + dxi and 

 yi -5- dyi, as also appears on putting x = Xi^ ndx-i and y = yi -f ndy^, and 

 using Euler's theorem on homogeneous functions. 



The vector expression for the oscnlating plane at the same point 

 is, if u, V, and to are variable parameters ("whose ratios only are 

 essential), 



P = 



^d^^'-'d^r''dij;y^'^'^^ 



d? d' d'\., ' 



dx,- dx.dy, dy^j' ' '^'^ 



Retaining only terms of the second order, it is obviously possible 

 to expand 



(^ [x^ + dx^ -f ^d-x^, yi + dy^ + ^d'-y^) 

 in the form 



/' ^ cT- ^ dr-\ 



in -which u^, i\, and Wi are independent of the coefficients of A, and 

 involve only Xi, yi, their deriveds, and the nxunber n which determines 

 the order of the binary (}>{xy). This being so, the vector expression 

 lately written involving linearly two independent parameters (the 

 ratios of u, v, and tc), is seen to represent the osculating plane at Xjyi, 

 as in the neighbourhood of the point the deviation of the curve from 

 the plane is a quantity of the third order. 



K.I. A. PEOC.. "^ER. m., VOL. IV, 2D 



