AND DIAGRAMS OF FORCES. 185 



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

 FF,_, then we must combine equation (6) with 



-F 1 ............................ (7), 



whence eliminating <, 



If r, 2 is the length of the line drawn from the first point P t to the second 

 P t ; and if l u m u n n are its direction cosines, this equation becomes 



IJi+'mtf + nJL = -^ ' , 



'13 



or the reciprocal of the two points P l 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 PjP., is equal to the excess of F n over F } . 



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

 are x 3 , y t , z, and for which F=F S ; then we must combine with equations (6) 

 and (7) 



F 3 ............................ (8). 



The reciprocal of the three points P,, P 2 , P, is a straight line perpen- 

 dicular to the plane of the three points, and such that the perpendicular on 

 this line from the origin represents, in direction and magnitude, the rate of 

 most rapid increase of F in the plane PfJP^, F being a linear function of the 

 co-ordinates whose values at the three points are those given. 



(4) Let there be a fourth point P 4 for which F=F t . 



The reciprocal of the four points is a single point, and the line drawn 

 from the origin to this point represents, in direction and magnitude, the rate 

 of greatest increase of F, supposing F such a linear function of xyz that its 

 values at the four points are those given. The value of <f> at this point is 

 that of F at the origin. 



Let us next suppose that the value of F is continuous, that is, that F 

 does not vary by a finite quantity when the co-ordinates vary by infinitesimal 

 quantities, but that the form of the function < is discontinuous, being a different 

 linear function of xyz in different parts of space, bounded by definite surfaces. 



VOL. II. 24 



