170 ROCHESTER ACADEMY OF SCIENCE. [Jan. 14, 



In the planar product of a point and two posited vectors, we get, 

 as might be expected, a mixed product, regressiv^e from the product 

 of the two posited vectors, and combinatory from the result of the 

 point into the regressive point, viz. : 



pL,L,=p -p^p, ■p,p„_=p -A ■P.P.P.=PPo [P.P. P. 



= a posited vector with a weight A^i A- 



The product of three coplanar posited vectors : 



L^ L, L, = L, . L, L, = /., . p, . ?np^p,p, 



= L,p, ■ mp^p^p, 



= l^P.P.P, ' ^^AAA = ^/3(AAA)' 

 m, /9, being scalars, and /,, />.,, p^ the points of intersection of the 

 posited vectors. 



IX. 



Two points are independent of each other, that is, one cannot be 

 expressed in terms of the other, whereas of three collinear points, any 

 one can always by a proper adjustment of weights be considered as 

 the mean point of the other two, that is, is pendable on them, to coin 

 a word for the moment. 



Of three points at random, no one can be made the mean of the 

 other two, and, therefore, three points are independent. If a fourth 

 point be introduced into the plane of the three, it can by a proper 

 adjustment of weights be made a mean of the other three, and, there- 

 fore, four or more co-planar points are pendable. Similarly four 

 points in space are independent ; five or tnore points in space are not 

 independent. Two non parallel vectors are independent ; two parallel 

 vectors are not independent. Three vectors not parallel to a plane 

 are independent, but not independent when parallel to a plane. Four 

 or more vectors are not independent. We have found the product of 

 two points to be the posited vector between them. If we attempt to 

 multiply by a new point collinear with these, the combinatory product 

 is nil. Similarly with three vectors in a plane, and so on. This sug- 

 gests that \\\^ product of pendable factors is nil, which will be found to 

 hold for four coplanar points, three coplanar vectors, etc., unless the 

 product of previous factors (counting from the right as written) 

 becomes regressive or scalar. For example, in the product of the 

 three coplanar posited vectors A, L„ L^ of section VIII. 



Hence the laws of section VII. should have been restricted to 

 independent factors. 



