86 MASS AND FORCE. [CHAP. V. 



68. Material Figure. Mean Acceleration. Imagine that 

 all the points which are within a certain volume at time t define 

 particles (Art. 41) which are moving relatively to a certain frame. 

 The figure constructed with points in the positions of these 

 particles at any subsequent time is called a material figure* 

 provided the accelerations of the points are subject to certain 

 rules to be expressed in the two following Articles. 



Whatever these accelerations may be we can define, as the 

 mean acceleration of the figure at time t, a vector, localised in a 

 line through the centroid of the figure, whose resolved part in 

 any direction is equal to the average acceleration per unit volume 

 of the points of the figure in that direction. 



Thus if x, y, z are the coordinates of one point of the figure 

 at time t, dv an infinitesimal volume containing the point, and v 

 the volume of the figure at time t, the resolved parts of the 

 mean acceleration of the figure parallel to the axes are 



the integrations extending through the volume. 



It is clear that the mean acceleration at any instant is the 

 acceleration of the centroid of the figure at that instant. 



The rules, above mentioned, to which the accelerations of the 

 points of the figure are subjected, are rules governing the magni 

 tudes, directions, and senses of the mean accelerations of two or 

 more material figures. 



69. Mutual Action. Imagine that the positions (or velo 

 cities) of two material figures affect their mean accelerations 

 relative to the same frame; in other words that there is some 

 relation between these accelerations and the positions (or 

 velocities) of points in the two figures. Then we say that one 

 of them acts on the other and produces acceleration in it. 



Imagine that two material figures act on each other so that 

 each produces in the other a mean acceleration relative to a 

 certain frame. Let/,/ be the magnitudes of these accelerations. 



Now we subject these mean accelerations to a rule, viz. they 

 are localised in the same line, the line joining the centroids, and 

 have opposite senses. 



* Cf. Maggi, Teoria matematica del movimento dei corpi. Milan, 1896. 



