138 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



Hence the total energy is obtained by adding to the right-hand 

 side of (21) an expression of the form 



where 



121. The impulsive forces necessary to produce from rest 

 the actual motion at any instant now consist partly of impulsive 

 forces applied to the solid, and partly (as explained in Art. 61) 

 of impulsive pressures pic lt pte z , &c. uniform over the several 

 membranes which are supposed for a moment to occupy the 

 positions of the barriers above-mentioned. The components of 

 the force- and couple-resultants of the first set, we denote by f t , 

 rj lt fi, and X x , /j. lt v lt respectively; those of the force and couple 

 equivalent to the second set by f 2 , rj 2 , f 2 , and \, p 2 , v 2 . By the 

 impulse of the motion at any instant we shall understand the 

 force and couple equivalent to both these sets combined, so that 

 if ?&amp;gt; V) ?J \ P* v b e ^ s components, we have 



f = + &&amp;gt; &c., &c, 

 \ = \+\ t &c., &c. 



If we use the term impulse in this sense, the reasoning of 

 Art. 108 and consequently the equations of motion (20) will still 

 hold. The formulae (23), however, connecting f, y, f, &c. with T 

 require correction. 



By the same reasoning, and with the same notation as in 

 Art. 112, we have 



+PX.+ +*,,+ ) 



