20 MR CLERK MAXWELL ON 



Substituting the values of £, 77, £ from (1) 



dd> dF dx dF dy dF dz dF 



dg dx dg dy d^ dz d% d% 



dF dF 



= x 

 Differentiating <f> with respect to -q and £, we get the three equations 



x - d% y - d V z - d$ • (°>> 



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

 magnitude the rate of increase of </> 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 Q z 



f = f. + c_o fo + (y _ yo) <| + ( ,_ 2o) a, (4) 



The value of F 7 at the origin is found by putting x t y and z — 



F = F - x£ - y oV - z£ = - tf> . . . . (5), 



or the value of F' at the origin is equal and opposite to the value of 4> at the 

 point £ 7), I 



If the rate of variation of F is nowhere infinite, the co-ordinates $ 77 £ of the 

 second diagram must be everywhere finite, and vice versa. Beyond the limits 

 of the second diagram the values of x, y, z, in terms of £ 77, £, must be impossible, 

 and therefore the value of <j) is also impossible. Within the limits of the second 

 diagram, the function <f> has an even number of values at every point, corre- 

 sponding to an even number of points in the first diagram, which correspond 

 to a single point in the second. 



To find these points in the first diagram, let p be the vector of a given point 

 in the second diagram, and let surfaces be drawn in the first diagram for which 

 F is constant, and let points be found in each of these surfaces at which the 

 tangent plane is perpendicular to p, these points will form one or more curves, 

 which must be either closed or infinite, and the points on these curves corres- 



