DYNAMICS. 39 



i 



For as the quantities of motion before and after colli- 

 sion are the same, (73.) 



A a -|- B b _= A of + B l/ 9 

 or A(a a') = B (^ 5); 

 and since (79) <* 4- a 7 = 6' -t- b, 

 therefore A (a 2 V s ) = B (&* *), 

 or Afl* + B6- Aa" + B b*. 



See another demonstration, in MACI-ACRIX'S Account 

 of NEWT ox's Discoveries, Book n. chap. iv. 12. 



81. If between two unequal elastic bodies A and 

 C, a third B be interposed ; and if the least A, be 

 made to strike with any given velocity on B, the 

 motion communicated to C will be the greatest 

 poble, when B k a mean proportional between A 

 andC. 



It is easily shewn, from 78. that the velocity com- 

 municated to C is = 



4ABa 4Afl 



(A + B) (B+C) 



This fraction is a maximum, when the denominator 



1C 



A -f B -h \ - -f C is a minimum, that is, since 

 B 



A and C are given, when B 5 AC, or when B is a 

 mean proportional between A and C. 



Tbe 



