IO 



Ellery IV. Davis 



p' -f- p." These divisions are represented in fig. io by P 3 and 

 P i respectively. Reflect that the black line-segment represents 



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Fig. io. 



the growth in the black vector, as keeping real, i. e., black, it 

 changes from the black vector of P x to that of P 2 , while the blue 

 line-Segment represents the corresponding growth in the double- 

 vector, the sum of the growths of the constituent vectors. 



When instead of division in the ratio of q' -\- q" : p' -\- p" we 

 have that in the ratio q : p each of these growths is divided i : n 

 that ratio. Thus, instead of the black growth from P 3 to P 4 we 

 have a red-growth which will therefore change the red vector, 

 while instead of the red change in the vector from that of P 3 to 

 that of F 4 we shall have an opposite black change. The result- 

 ant red vector is P 5 . In the figure the vectors marked a are each 

 equal to P 3 — P 4 while P n is equal to P, -f- b. 



Any point in the plane is the black end of a red vector dividing 

 the line connecting two vectors in some ratio or other. For let 

 the vectors be P ly P 2 . Join their black points by a black line 



Fig. ii. 



and their blue points by a blue line, and from the arbitrary point 

 in the plane draw a black vector in the direction P 1 — P 2 , or its 



io 



