THE LAWS OF MOTION 319 



length (p. 304). If B and C do not modify each other's 

 influence, then their combined action, given by the ac- 

 celeration d, corresponds to a force which, measured by the 

 product of mass and acceleration, or by m x d, is in times 

 the step d. This force is termed the resultant force ; and 

 we see that, since the resultant and component forces are 

 respectively iu times the diagonal and the sides of the 

 acceleration -parallelogram, these forces must themselves 

 form the diagonal and sides of a parallelogram A /3 B y 

 which is a magnified picture of the acceleration-parallelo- 

 gram. This is the famous parallelogram of forces^ and we 

 notice that it follows at once from the parallelogram of 

 accelerations when we assume that B and C do not modify 

 each other's action/ 



If they do modify each other's action there will still be 

 a parallelogram (A ^' S' y) of forces, namely, the resultant 

 force iu X a^ will be the diagonal of the parallelogram on 

 the sides m x d^ and m x c'. But if we mean, as physicists 

 generally do, by the force of B on A the force when A 

 and B are alone in the field, and similarly by the force of 

 C on A the force when A and C are alone in the field, 

 then we must assert that on the hypothesis of modified 

 action : T/ie parallelogram of forces is not a synthesis by 

 which we can truly combine forces. 



This conclusion may appear to the reader so entirely 

 opposed to all that he has read in text-books of mechanics, 

 that he may be led at once to reject the hypothesis of 

 modified action. One of Newton's laws of motion dis- 

 tinctly excludes indeed this hypothesis, and a great 

 simplification in our process of constructing complex from 

 simple mechanical systems undoubtedly arises when we 

 exclude it ; we have not to deal with every new field 

 afresh, and to re-measure accelerations for each variation 

 of its constituent elements : we simply analyse it, break 

 it up into simple fields, the individual motions of which 

 have been previously discussed. Yet it is not scientific 

 to assert that the simplest hypothesis is necessarily correct 



1 This, for the purposes of the physics of the particle, might be spoken of 

 as the seventh law of motion. 



