COMPOSITION OR AGGREGATION OF FORCES IS A CONSEQUENCE. 303 



f 



which is a part of our temperament. This principle would justify our declaring, as a result 

 of thought, that in the same or equal masses, accelerations are as the pressures which produce 

 them. Now here we know and measure pressure by means totally different from those by 

 which we know and measure acceleration. We find that pressures are as the produced accele- 

 rations: but we can think of them as under other laws. 



Should the pressures not be as the velocities produced in given times, one of two results 

 of physics is false : either the fourth postulate is not true of pressures, or the velocity due to 

 the aggregate is not the aggregate of the velocities due to the aggregants. If v and w be two 

 velocities due to two pressures (pv and (pw, in the same direction, and if (p(v + w) be the 

 pressure due to the aggregate velocity, then if (pv + (pw be the aggregate pressure, and v + w 

 its velocity, we have (pv + (pw = (p(v +v}), which gives 0» = c«, if v and w be wholly unre- 

 lated to each other. That pressures are as velocities produced, important and fertile as the 

 principle may be, is not a fundamental law in our order of thought. Dismissing the pure 

 result of thought that velocities are aggregated by addition, the connexion of pressure and 

 velocity depends on, and is a mathematical consequence of, two more simple laws. First, 

 that pressures producing motion in one direction are aggregated by addition : secondly, that 

 the velocity due to the aggregate is the aggregate of the velocities due to the aggregants. 

 The question now is, do these two laws, when the first three postulates are introduced, show 

 any dependence in thought each on the other. 



The first of these two laws, which is in fact the fourth postulate, joined with the other 

 three, gives the diagonal law of aggregation to pressures producing motion : that is, the 

 pressures are aggregated by the same law as the velocities. But this deduction, extending as 

 it does to all combinations of direction, does not advance us one step towards the conclusion 

 that the velocity due to the aggregate of pressures is the aggregate of the velocities due to the 

 separate pressures: the diagonal pressure does not necessarily produce the diagonal velocity. 

 If (pv be the pressure which produces in a given time the velocity v, then v and w, imparted 

 at an angle 9, give ^(v'^ + w^+2vwcos9) in the diagonal: while ^i/ and ^w produce the 

 pressure -^{{(pvy -t- {(pwf + 2(pv . <pw .cos 9 \ in the diagonal. The equation 



^{^{v^ + w'' + 2vwcos9)] =\/{{(p'vy + ((piv)- + 2(pv . (pw . cos 9 \ 



may not be satisfied : nor can we deduce it from any simple law in addition to those given, 

 except ^v + <pw = (p(v + w). 



Laplace {Mec. Cel. 1. i. c. 6) discussed the effect which would be produced upon the equa- 

 tions of motion by the assumption of (pv left indeterminate. In this discussion, however, he 

 assumed the diagonal law of aggregation of pressures producing motion : for his aggregants of 



dx 

 (pv are (pv — , &c. This was but a partial inroad into the region of physical impossibility: 



CLS 



he ought to have left the law of aggregation of pressures as open as the law of connexion of 

 pressures and velocities; but all his unnatural conduct consisted of dispensing with the 

 principle that the sum of the causes is the cause of the sum. 



Vol. X. Part II. 39 



