A SYSTEM OF PARTICLES. 273 



from which it appears that f{x) = 0. or m is a constant = V; and 

 hence a = Vt \ ffe"^'.cos*^ is a solution of the equation. 



40. ^^^e conclude that a motion of translation is perfectly con- 

 sistent with vibrations, and, from the form of tlie solution, it will be 

 perceived that the transmission is not uniform, but proceeds as it were 

 by fits; the uniform motion of transmission being combined with a 

 variable one, depending on the lengths of the waves in the trans' 

 versal vibration. In other words, at particular points the direct motion 

 IS accelerated or retarded by the eflfect of the transverse motion of 

 the particle. 



41. In order to ascertain the value of V it will be necessary to 

 recur to the circumstances under which any particle started into motion. 



Let the particles behind it be in motion according to the regular 

 type, then clearly the force acting on this particle at the first moment 

 is represented by 



2 ^-^ + « „lx 



taken only on one side of it with reference to a plane perpendicular 

 to the axis of x. 



The expression becomes by expansion. 



^ ^^ + « ll _ 3 2 {a^X + ^hj + y^%) + a^ + ^a ^ y 



r* ■( 2" ~ ^5 



_l_ 3^ 4 {a'hx-'- + 0'Sf + y^s') + 8 {SxSijul3 + ...)| S c 



2.4 r' j - ^-pr 



_ V -^ / ^ 2 (a'o.r + afivl/ -|- aySx) 3 



" r' r 2 • f^ 2 r'- ^•^' ^^' + "^'^ 



^3^ 4 {^x:' . «' + ^xhf . /3° + ^x^z'y-) 

 2.4 ' 'r^ 



3 2 . a^x' + {a" + 0,' + y') ^x 



2" ? + 



•■) 



