348 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Consider a variety Ts traced on $; that is, let the coordinates .r be 

 functions of the five parameters Ui, Ui . . . u^. Here and following 

 all the functions used are supposed to be continuous and to possess all 

 partial derivatives of the ;?'*» order. The tangent lines to V^ will 

 generate an aSj determined by the six points ^ 



^""^^ '^''^' \j^J' \dirj' [du,]' Va^J' \dlfj- 



Through this S^ will pass oo* hyperplanes which are tangent likewise 

 to 1^5 (an hyperplane is said to be tangent to a variety if it contains 

 the tangent space to the variety). Hence, In each line of a five- 

 parameter family Vs there are oc* C;,'s tangent to it. 



The osculating planes of all the curves traced on F5 which have the 

 same tangent line in x generate an *% which contains the space ttj 

 tangent to Tg, as may be seen as follows : 



Using Segre's*^ notation, the osculating planes will be determined 

 by the three points 



(4) f= 0, 



(5) ^fdui = 0, 



(6) 1fi„dUiduic + ^fd^Ui = 0. 



Tangent lines are lines which join (4) to any point of the space deter- 

 mined by /i, f, fa, fi, fh and hence generate ttj. Now, if the du^ are 

 held fixed (that is, the direction of the tangent is fixed) and the d'Ui are 

 allowed to vary, we see that the osculating plane will always stand in 

 the /Se defined by 



^ A tangent line joins the points x and x + dx, or expanding x and 



^\dx 

 X + Zi-T—diH. As the dux determine the direction along Fs, the space 



generated by these lines will be the /Sj named. 



' "Su una classa di superficie ecc." Atti di Torino, 1907. Points are 

 represented by their equations in hyperplanar coordinates or also with the 

 first member of these equations. Thus 



where iS^ are the hyperplanar coordinates. The point which describes a 

 variety shall be represented bj' /. We may also speak of the point x, repre- 

 sented by /, or S^Wx^*) = 0. Subscripts are reserved to denote derivatives; 

 thus, 



^^^ duidujduk 



