13-2 Froceedinffs oj the Royal Irish Academy. 



Mr. ^V. E. Eoberts has giveii expressious for the coordinates of a point on 

 UV of the tangent line and osculating plane. (Salmon, Analytic Geometry 

 of Three Diinensmis, p. 226). In tl.e present notation we have, in general, 



I^m..^na^^ (1) 



/(a) 

 whatever I, m, n may be, where /(«) is the residt of substituting the value a 

 in the derived function of /(A) 

 Hence the values 



,A-a '_^~^ -_ ^ ~ 7^ -T ^ A - o 



^^m' ^'fWr ""/(t)' ''"^/(2) 



satisfy the equations r= 0, F= for all values of A, and 



(2) 



^Jfd'^' 



are the coordinates of a poiut on UV. 



A generator of one system of XU - V may be regarded as the inter- 

 section of the planes 



- 6 jAa . X-h iOjXs .W -5- JA/8 . y -5- iJXy . i = 0, 



JXa-x^ iJXs.w + dJXB.y "iSjXy .z = 0. (3) 



Forming the line coordinates in the usual way, and dividing out by - i, 



p = 20 Ja^s, 2 = (1 - e--)jA^s, r = i (1 -f r-jjAVs. (3a) 



s = 2eJX~^, t = a-0')JXy., u = iil + e'-)JXa0. 

 A generator of the opposite system is the intereection of the planes 

 ~ 'PjXa.x- i4J,JX5.'m-tJXfi.y-i-iJXy.z = 0, 

 JAo . « - ij>^ . fr -f ^ JXp . y - iipJXy . ^ = 0, 

 and the coordinates of this generator are 



/ = -29JAa5, 2' = -(l-^'-)JA^s, r'.= -i(l4-^'-)jA;5. (4) 



/ = 2,iJA^, t' = (1 - <l>^)JXya, r' = t(l + ^'-)jA;^. (4a) 



Hence, the coordinates of any two generators of the same system are 

 connected by the relations 



ps' - p's = 0, qt' - q't = 0, ru - r'u = 0, (5) 



while those of opposite systems are connected by the relations 



jps -f p's = 0, qt' ^ qt = 0, ru ^ r'u = 0. (6J 



Sohing from the equations (3), (4], the coordiaates of the intersection of 



