RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 21 



pond to the points in the second diagram which lie in the direction of the 

 vector p. If p be the perpendicular from a point in the first diagram on a 

 plane through the origin perpendicular to p, then all those points on these curves 



at which -r- = p correspond to the given point in the second diagram. Now, 



since this point is within the second diagram, there are values of p both greater 



and less than the given one; and therefore -y- is neither an absolute maximum 



nor an absolute minimum value. Hence there are in general an even number 

 of points on the curve or curves which correspond to the given point. Some of 

 these points may coincide, but at least two of them must be different, unless 

 the given point is at the limit of the second diagram. 



Let us now consider the two reciprocal diagrams with their functions, and 

 ascertain in what the geometrical nature of their reciprocity consists. 



(1.) Let the first diagram be simply the point P a , {x 1 ,y v zj, at which F = F 1 , 

 then in the other diagram 



<j) = x^ + y lV +e l C-F 1 (6), 



or a point in one diagram is reciprocal to a space in the other, in which the 

 function </> is a linear function of the co-ordinates. 



(2.) Let the first diagram contain a second point P 2 , {x v y v z 2 ) at which F = F 2 , 

 then we must combine equation (6) with 



= ^ + 1/ 2 V + *<£- F 2 • • • ■ (7), 



whence eliminating <f>, 



(«i-a^)£+ fa-yjv + (%-z 2 )f = F : -F 2 . 



If r 12 is the length of the line drawn from the first point P x to the second P 2 ; 

 and if l 12 m 12 n 12 are its direction cosines, this equation becomes 



F — F 



*ia£ + m i2*? + n uZ = -*z — > 



'12 



or the reciprocal of the two points P x and P 2 is a plane, perpendicular to the line 

 joining them, and such that the perpendicular from the origin on the plane 

 multiplied by the length of the line P X P 2 is equal to the excess of F 2 over F^ 



(3.) Let there be a third point P 3 in the first diagram, whose co-ordinates are 

 # 3 y z z 3 and for which F = F 3 ; then we must combine with equations (6) and (7) 



</> = a' 8 £ + VaV + z*K- F 3 .... (8). 



The reciprocal of the three points P a P 2 P 3 is a straight line perpendicular to 

 the plane of the three points, and such that the perpendicular on this fine from 

 the origin represents, in direction and magnitude, the rate of most rapid increase 

 of F in the plane I t 1 P 2 P 3 , F being a linear function of the co-ordinates whose 

 values at the three points are those given. 



VOL. XXVI. PART I. F 



