JoLY — Vector Expressions for Curves. 393 



notation, suppose the degree of the curve to be m. The scalar equa- 

 tion of the osculating plane is 



\dx- ' dxdi/ dif dxdy ' dif- dx' d\f ' dx- dxdy ] 



d'cfi d-(f> dr(ji 

 ' dx^ dxdy dy"^ ' 



and as this inrolyes x : y in. the degree 3 («» - 2), the number of oscu- 

 lating planes through an arbitrary point is w = 3(m - 2). 



In like manner, if the tangent (in which x' and y' are variable) 



, d<^ , d(fi 



x'j^ + y'^ 



_ dx dy 



dx dy 

 meets an arbitrary line p = a + t^, 



\dx ' dy dy ' dxj dx dy ' 



and as this involves x : y \n. the degree 2{jn - 1), the rank of the 

 curve is r = 2(;«- 1). From these three all the characteristics may 

 be deduced. 



It is simpler, perhaps, to notice that the curve is the reciprocal 

 with respect to the sphere p^ + 1 = of the plane 



Sp<i,{xy)+f{^xy) = 0, 



which involves the parameter x-.yin. the degree m. The characteristics 

 of the curve are thus the reciprocals of those given in Art. 329 of the 

 "Three Dimensions." They are, in Dr. Salmon's notation, 



a = 4(w-3); x = 2{m-\){ni-2>) \ h = i{m-l){m-2) ; 

 ^ = 0; y^2{m-2){m-3); ff = i{9>n--o3m+80). 



In Art. 349 it is shown that the quartic considered in Art. 17 of 

 the present Paper is the excubo-quartic through which only one 

 quadric surface can be drawn, 



2E 2 



