6 SIXTH REPORT. 1836. 



US to this double value is obvious when we consider that all perpendi- 

 culars upon one side or face of the given line being considered as posi- 

 tive, all on the opposite side or face must be marked as negative; for if 

 the given line be supposed to revolve in the plane of the coordinates, 

 any point of it being fixed as soon as it has passed through a semirevo- 

 lution, it will teike a position in which the very same equation as at first 

 will belong to it, and in wliich the perpendicular upon it from the 

 given point p will have exactly the same length ; and indeed be the 

 very same line that was perpendicular to it in its first position. 

 In the first position the perpendicular from P falls upon the face of 

 the line which is then turned towards it ; but after the semirevolu- 

 tion, the perpendicular from P falls upon the face of the line which, 

 in its first position had been averted from P ; and hence one of 

 these perpendiculars is presented to us by the analytic investigation, 



y jj ^ J) 



as + <- — ■ while the other from the same point, P, upon 



Vl+a^ 



the line expressed still by the same equation, y — ax — J = o, is 



y' — ax' — h 

 brought under our notice, as, — — , . 



That this is the true origin of the double sign found in the in- 

 vestigation of the length of the perpendicular, will be still more 

 clearly seen by tracing the varying length of the perpendicular, as 

 the line D^ C D revolves from its first position. Let us suppose, 

 (in order to fix our ideas) round some point, as C, which we may 

 suppose to hold its place. Then as the revolving line approaches 

 P the length of the perpendicular diminishes ; when it reaches P 

 that length vanishes ; when it passes P the perpendicular now 

 falling on the face that had been at first averted from P, becomes 

 negative ; or, rather, has a sign opposite to that which we first at- 

 tributed to it ; and this sign it retains as long as the perpendicular 

 continues to fall on the same face ; and therefore, when it has passed 

 through its semirevolution, it retains that contrary sign ; but at the 

 end of the semirevolution, the perpendicular is the very same as it was 

 at first, and the line in the new position has the same equation that at 

 first belonged to it ; the face alone on which that perpendicular falls 

 has changed, and algebra marks that change by the change of sign 

 of the value of the peri^endicular. Indeed, it is easy to see that 

 after the semirevolution is completed, a perpendicular P* A^ at an 

 equal distance from C\ and similarly situated on the other side 

 from P A, and erected upon the opposite face of the given line, 

 will have come round to the portion of P A, and will then coincide 

 with it, if it be supposed to accompany the revolving line, and to be 

 inflexibly attached to it. 



An account nearly the same can be given of the double sign of 

 the distance between two points (.r\v') and (.^^^V")' which, as is well 

 known, is = -f i/(x^ — ^^*)'- + (y^— y")-. If we at first arbitrarily as- 

 sume the + value as belonging to the distance : then if a point be 

 made to move from that given point, which is nearest to the origin of 



