347 



^^o / > i ^ t* \ i f i7 sin(E,H-tei— /to)) 



r os ( t i+^i-^-)+j_ oo ^ ^=^ i>\ 



d/ 



that is, nearly, if a- be considerably different from ^^yy, 



v,< = * x \ A Vi cos ( £ \ + ts \ - **) 



We have therefore the approximate expressions : 



(21) 



(21)' 



%,k,t = x \ *\,i cos ( E \ + M - **)• if * <*^ S (22) 



and 



-To* 



S^, = 0,ifi>^ ! (22)" 



we have also nearly, in general, 



-/„* 



but the discussion of the case when a; is nearly zz^-^-is too 

 long to be cited here. The formula (22) for £ , coin- 



cides with the particular integral (4) v ; and the condition 

 which it involves with respect to x, expresses the law ac- 

 cording to which this particular integral comes to be 

 (nearly) true for greater and greater positive values of x and 



ds 

 t, (if -tt^>0,) after having been true only for negative values 



of x when t was — 0. 



In the particular case n = 8, the foregoing formulae 

 have an immediate dynamical application, and correspond 

 to the propagation of vibratory motion through a system of 

 mutually attracting or repelling particles ; and they conduct 



