76 Froceedings of the Royal Irish Academy. 



Two planes, of course, may not have a line common to both ; for 

 instance, the plane of «\ and i^, and that of 4 find 4 have no common 

 line ; hut every line in one of these planes is perpendicular to every 

 line in the other. 



In like manner, the angle between two spaces of three dimensions 

 having a common ^^^ane is the angle between a pair of lines, one in each 

 space, and both of which are perpendicular to the common plane. 

 Two consecutive spaces osculating to a curve have common an oscu- 

 lating plane. The angle a^ds is the angle between the perpendiculars- 

 in these spaces to that common plane. 



Three mutually rectangular lines (e'l, u, and i^) determine a space 

 of three dimensions containing all the vectors x^ii + Xoi^ + x-^iz. The 

 vectors «'i, 4? and 4 cos 9 + i^ sin 6 (which are also mutually rectangular) 

 determine a second space having the plane of ii and fe common to both. 

 In the first space the pei-pendicular to this plane is 4, and in the second 

 it is «3 cos 6 + ii sin 6*, and is the angle between these lines, or the 

 angle between these spaces. 



Of course two tri-dimensional spaces may have only a line or a point 

 common, or no point may be common to both. Two such spaces must 

 have a common plane when both are contained within a space of four 

 dimensions ; a common line, when a space of five dimensions contains 

 them ; a common point, when the containing space has six dimensions. 

 In a space of seven dimensions, the spaces represented by 



p - Xiii + Xoio + x^iz and p = a~,i-, + x^i^ + x^i^ + x^i^ 



have no point unless the constant a-, happens to be zero, 



5. Continuing the process of the 3rd Article, it is found that 

 Dstti = a^a.^ - a^az, and in general that X)sa„,_i = ff„i-ia,„ - a,n-20-m-ir 



until all the independent vectors are exhausted. At last, if the curve- 

 is contained in a space of w dimensions, Dja,, = — «„_ia„_i. 



In terms of the n-1 scalars ai, a^, . . . «„_i, all the affections of the 

 curve can be expressed. Differentials of any order of the vector p to 

 a point on the curve, and of any of the derived vectors a, may be 

 reduced to linear functions of the a with scalar coefficients composed 

 of the scalars a and of their differentials. 



6, These formulae may all be collected into a single type expressed 

 by the equation D^a^ = ViQa,„, in which O is a sum of binary products 



