g^ On Machines in General. 



rections, bnt ^v ill merely have becomes times as great a. 

 hev ";ald have been if the bodies bad been hard : th. 

 bein. done, since W i. the result of V and U, we have V 

 cosine Z = W cosine Y - U : thus ihe equation (E), for 

 which wc are seeking one analogous, may be put under this 

 form 5 m W U co.:ne Y - . m U' ^ 0. Now, according 

 to what has been said, we must, in order to apply this equa- 

 tion to the case in question, place ^ in place of U, without 

 any chancre r.pon Y: therefore in the case we are examining 

 the equation wUl be 5 m V/ - cosine Y - 5 -— - ; or 



bv multiplying by.S n..AVU ^f-^^'l'^^'^'^ll 

 or on account of W cosine Y :=^ V cosine Z + U, we 



shall have ^ .»VU cosine Z == s m U- thus this 



equation will be, with respect to the bodies in question 

 what the equation (E) is with respect to hard bodies; and 

 even the latter is the particular case where we have « = 1, 



^' When'n =2, it is the case of bodies perfectly elastic, and 

 the equation becomes 2.mVU cosine Z + . ™ U^ = O 5 

 but this equation relative to bodies pertectly elastic may be 

 expressed in a known and mu.e simple manner, as follows : 

 Since W is the result of V and U, we have by tngonornetry 

 YJ^^_ V^ + U^ ^ sVUcosineZ; and therefore 5 TO W^^ 

 smV^ + ^-"^U^ + 2 ^^nV\J cosine Z. Adding to this 

 equation that found above, and reducing, we have 5 to W 

 = ,TO V^ which is precisely the principle of the preserva- 

 tion of active forces, i. e. this preservation is, with respect 

 to perfectly elastic b-.dies, what the equation (E) is with 

 respect to hard bodies, as we undertook to prove. 



First Remark. 

 XXVII. I shall noi dwell on the particular conse- 

 Quences which I miffht draw from the solution of the pre- 

 ceding problem ; but^shall merely remark, that the velocities 

 vv V U, being ahNav3 in proportion to the three sides 



