206 Mr. JMallet on the Physical Conditions 



the equation then becomes 



^1±Jl1v"'^2{1' + 1") {L'-l') + r- = L"; (54) 



and abridging, as before, by making 



k' = 



l' + l"J~yl\{P + F)Lj' 



we have 



L'^LI=l'+r'±J{l"'+^]; (55) 



twice which l(l"' + -pT:-,)±(i' + l"), corresponds to the maximum elongation 



or compression, and the smaller value to its minimum, if the sign be +, or to its 

 maximum if it be — , that is to say, according as P' acts with or against P, 

 assuming always that M, M\ and L remain in contact after the impulse 

 of if'. 



127. The fundamental equation of motion, by which the velocity V, common 

 to 31 and J/' at any point, where I = Li, is obtained on the principle of vis viva. 

 For the increment of vis viva of J/ and M', since the commencement of motion 

 (M+M') (V- V"-) is double the work {P+ F) {I -I') developed in them 

 and due to the H' = 1- /', less by twice the work, 



le4i(r-i'^) = r-^(^— n, 



which is thus developed in the opposite direction by the elastic resistance of 

 the bar eAi, during the time due to H' = 1 — I' . 

 To obtain F, therefore, by means of /, we have, 



{P±P)_y. _{l±Illv''^^'L {p ^F) {i-v) -~ (I' -r-\ 



9 9 ^ 



multiplying all the terms by —^, 



^ - ^ = 2 (/' - Z") {I-}!)- r- + P = I'" - (/' + r - If. 



