AND DIAGRAMS OF FORCES. 183 



Next, let us determine another function, <, from the equation 



(2), 



$, as thus determined, will be a function of x, y, and z, since 17, are 

 known in terms of these quantities. But, for the same reason, <f> is a function 

 of 17, . Differentiate < with respect to , considering x, y, and z functions 

 of r?, , 



dxdydz dF 



_ . 

 d~ 



Substituting the values of 77, from (l) 



d<k dFdx dF dy dF dz dF 



_ II I* -L _ _ I _ *?, l 



dg * dx d^ dy d + dz dg d 



_ 



*d 

 = x. 

 Differentiating (f> with respect to 77 and , we get the three equations 



~' ~' ~ 



or the vector (r) of any point in the first diagram represents in direction and 

 magnitude the rate of increase of <f> at the corresponding point of the second 

 diagram. 



Hence the first diagram may be determined from the second by the same 

 process that the second was determined from the first, and the two diagrams, 

 each with its own function, are reciprocal to each other. 



The relation (2) between the functions expresses that the sum of the func- 

 tions for two corresponding points is equal to the product of the distances of 

 these points from the origin multiplied by the cosine of the angle between the 

 directions of these distances. 



Both these functions must be of two dimensions in space. Let F' be a 

 linear function of xyz, which has the same value and rate of variation as F 

 has at the point x^y^ 



r.^+d.-^g+Or-rt^+ti-^J ............... (4 



The value of F' at the origin is found by putting x, y, and z = 



F = F t -x-yn-z=-4> ......................... (5), 



