1 66 ROCHESTER ACADEMY OF SCIENCE. [Jan. 1 4, 



ponents and at a distance inversely proportional to the weights of the 

 components, with its weight the sum of the weights of the components. 

 Accordingly if/>,,/>.„ wz,, m., designate the points and weights 



where p designates the mean point. 



This can be written m.,{p„ — p)^m^(^p — /J. 



The factors {p„ — p), {p — />,) are inversely proportional to the 

 weights of the components and, therefore, proportional to the dis- 

 tances of the mean point from the components, and, therefore, with 

 the proper unit of measure, equal to these distances. 



In />, + ^fip" ^ ( I + ^'^) py '^^ ^'^ decreases, the mean point p 

 approaches /,, coinciding with it for m = o, and finall}^ passes 

 beyond it to cc as /;/- approaches — i, and the equation becomes 



p, —p, = op 

 that is, the difference of two unit points is a point of zero weight at 00. 

 The meaning of a zero point at :« can be ascertained by considering 

 its effect. 



If/'cc denote a point at x 



mp^ +pzo 

 is a mean point of weight {m +1) lying in a determinate direction. 

 As VI increases, the mean point approaches p^, coinciding with it at 

 the limit 7n = cc, or dividing by ni = cc, p^ -\- o^» is a mean point 

 coincident at the limit with />,, but in a determinate direction of 

 approach, or, in other words, o/>cc has merely the effect of assigning 

 direction, that is, it is a vector, and, therefore, p^ — p,^-^op:^ is a 

 vector. The length of the vector /^ — /, is found from the equation 

 m„ {p., — p) = /«, {p — p^) to be equal to the distance between the 

 points, measured naturally from the subtrahend to the minuend. We 

 might have supposed a priori that the only difference between two 

 unit points would be difference of location, a vector, that which is 

 necessary to convert one point into the other. 



If p^ — /)„ =/>., — p^ ^ . . . . = op^, these vectors cannot be 

 posited ; their equality must be of magnitude and direction only — 

 excepting the limiting case of coUinearity. 



VII. 



Co7isidered as a symbol of operation, what is the potency of p^ on p^f 



The only difference between/., and/), is one of location, and the 



only way we can impress this difference of location on /, is to carry it 



