l68 ROCHESTER ACADEMY OF SCIENCE. [Jan. 14, 



the equal areas into two sets is by the adoption of the cyclical and 

 associative laws, viz. , 



P.P.P.=P.P.P.=P. -P.P-^P.P. ■ P. =■■■■ = — P.P. P. = 



From the analogy of \^ectors to velocities it is evident that — The 

 sum of two coplanar posited vectors is a vector passing through the com- 

 mon point of their extensions and equal in length and direction to the 

 resultant of the two vectors considered as velocities ; that is 



= 2A^4^=A(A+A) 

 /' 



Hence we have the distributive law for the multiplication of three 

 points. By reversing the direction of one of the vectors we get : — flic 

 difference of t7V0 co-initial posited vectors is a vector through the com- 

 mon point of their extensions a)id equal i)i length and directio7t to the 

 vector connecting their ends. 



For parallel vectors, the same modifications of these rules hold 

 as in the case of impulses, and the difference of two equal parallel 

 posited vectors becomes 



L^ — L., =/)j c — />„ £ ::= (/>j — p.f) I := o/'cc £ := a zero vcctor at cc. 



As in the case of the zero point at co^ we find that the zero vector 

 at '-r. becomes a mere plane direction, a plane vector. 



VIII. 



Ciuiding a posited vector by the characteristics of a posited vec- 

 tor, magnitude, direction and location of extension, gives a parallel- 

 opiped with the given vectors as edges. 



In plane space, since the posited vectors must have an element of 

 their extensions in common, viz., a point, we can write A, ^p^p^, 

 A^ ^p^^p,^. Of the products possible (all equal plane areas) by 

 different arrangements and associations, the two/>,,'s not being allowed 

 to come together in the same set, we find that the only possible basis of 

 division into two sets is the rule : — Looking from {or along) the guided 

 factor, if the points of the guiding (or guided) factor run in cyclical 

 order, or forioard 7vith the operand on the right, the product is 

 unchanged, and vice versa. 



