233 Cambridge Philosophical Society. 



the Composition or aggregation of Forces is a consequence." By 

 Prof. De Morgan. 



This paper examines the fundamental grounds of the composition 

 or, as Rlr. De Morgan calls it, aggregation of forces. By a tendency 

 is meant anything which has both magnitude and application : by ap- 

 plication is meant anj' notion which, not presenting the idea of mag- 

 nitude, presents the idea of opposition. Two tendencies ha\'e a third 

 tendency for their aggregate, to which they are jointly equivalent : and 

 equivalence is any notion which, given that things equivalent to the 

 same are equivalent to one another, satisfies the following postulates, 

 which are the grounds of every method of aggregation known in 

 mechanics. 



1. Any two tendencies have one aggregate (0, the aggregate of 

 counteraction being included among possible cases), and one only. 



2. The magnitude of the aggregate, and its application relatively 

 to the applications of the aggregants, depend only on the relative, 

 and not on the absolute, applications of the aggregants. 



3. The order in which tendencies are aggregated produces no 

 effect either on the magnitude or ajjplication of the aggregate. 



4. Tendencies of the same or opposite applications are aggregated 

 by the law of algebraical additions. 



From these postulates follow the following theorems : — 



5. In any aggregate, the result of partial aggregation may take 

 the place of its own aggregants. 



6. Two tendencies cannot counteract one another unless they 

 have equal magnitudes and opposite applications. 



7. An aggregate has not more than one pair of aggregants, when 

 the applications of the aggregants are given, and are different. 



8. If the aggregants be altered in any ratio, without change of 

 application, the aggregate is altered in the same ratio, also without 

 change of application. 



9. Any tendency may be disaggregated into two of any two dif- 

 ferent applications, neither of which is its own. 



From the preceding it is proved, — 



1 . When by application is meant direction, the law of aggregation 

 must be the well-known law of the aggregation of forces meeting at 

 a point. 



2. When by apjjlication is meant choice of a point through which 

 a given direction is to be drawn, the law of aggregation must be the 

 well-known law of aggregation of parallel forces. 



In the case of translations and rotations, the postulates are all 

 laws of thought ; in the case of pressures, whether divergent or 

 parallel, whether equilibrating or producing motion, all the postu- 

 lates contain results of experience. Accordingly, the multifarious 

 proofs of the laws of aggregation, in the case of pressures, are not 

 the mathematical playthings which they are often supposed to be 

 from their grounds being insufficiently stated. If to the postulates 

 necessary to make the law of aggregation a consequence, be added 

 the following, " the velocity due to the aggregate of pressures in one 

 given direction is the aggregate of the velocities due to the pressures 

 taken separately," it follows, as a mathematical consequence, that 



