JoLY — The Associative Algebra applicable to Hyperspace. 75 



As Sa^a^ = for all values of s, SaiD,a2 + SooB^ai = ; and hence D.a^ 

 lias the component - aiUi along aj. The new vector as is at right angles 

 to both tti and a^, and is the binomial ; ct^ds is the angle between as and 

 aa + BfL^ds, or the angle between the consecutive osculating planes, or 

 ^2 is the torsion. 



3. So far, all is the same as for three dimensions ; but the expression 

 ior D^az is different for the case of space of four and higher dimensions, 

 and for the case of three dimensions. 



Since Sa-^a^ — 0, SaiD^a^ + Sa^D^o'-i - 0- But the second term of 

 this differentiated expression is zero, because D^ai has no component 

 parallel to ag. It thus appears that SaiD^as = 0, and that D^a^ has no 

 component along ai. As in the last article, the component parallel to 

 tto may be shown to be - «2a2' There is no component along 0.3, and so 

 for three dimensions - an,a2 represents the whole vector. But, for four 

 dimensions, D^a^ may have, and in general will have, a component at 

 right angles to a], a2, and 03, or out of the space containing these three 

 vectors ; and if 04 is a unit vector along this component, 



4. In order to interpret the meaning of the scalar az a slight digres- 

 sion will be useful. 



A curve, unless it is a straight line, will deviate from a tangent. 

 Among the planes drawn through the tangent, one (the osculating 

 plane) will have the closest possible contact with the curve; but the 

 curve will deviate from the plane unless it is a plane curve. Among 

 the spaces of three dimensions that can be drawn to contain the oscu- 

 lating plane, one will fit closest to the curve, but it will contain it 

 only if the curve is tri-dimensional. The curve will in general deviate 

 from this osculating space. 



If a point moves along the curve with unit velocity, the tangent 

 line will turn round that point with an angular velocity equal to a-^ ; 

 the osculating plane will turn round that line with a velocity ffo ; the 

 osculating tri-dimensional space will turn round that plane with a velo- 

 ■city whose amount is a^. The angles between consecutive lines, planes, 

 •and spaces are, respectively, a^ds^ ((ids, and a^ds. 



In space of many dimensions the angle between two planes having 

 a common line is the angle between a pair of lines, one in each plane, 

 both of which are perpendicular to the line of intersection of the 

 planes. As each plane has many lines perpendicular to it, it will not 

 do to define the angle between two planes as being the angle between 

 the perpendiculars to the planes. 



