264 Prof. Powell's Abstract of M. Cuiichy's 



some more particular considerations have been lately sug- 

 gested as to certain conditions which are necessary to be ob- 

 served in order to the full application of M. Cauchy*s prin- 

 ciple. In fact, it becomes necessary to inquire into the more 

 particular nature of the relation he has established. To pro- 

 ceed, then, to this inquiry, we will resume the simple expres- 

 sion which gives the relation between the length of a wave 

 and the velocity of its propagation, viz. 



The particular nature of the relation depends on the value 

 of s; and for the inquiries to which we refer, it will be im- 



c 



portant to have before us the value of 5, or of ^, expressed 



in such a form as can be subjected to examination, so that we 

 may determine whether it fulfills the conditions presently to 

 be described. For this purpose, then, we must briefly deduce 



such a value of -p. 

 k 



The value of s^ depends upon the quantities L, M, &c., 

 which enter into the equation (29.); and the expressions which 

 these letters were assumed to represent are all of a similar 

 nature, consisting of the sums of products of several factors. 

 Taking a single term of the sum, these factors may be thus 

 simplified : 



From the assumption (22.) derived from the original equa- 

 tions (12.), we have 



r (2m fir) . 9 /^rcosSx) 



^ ~ I + { 2j«/Jr) cos' a sin' {^') } 

 which is easily reducible to 



L = 2 « L(r) + cos^ af(r) _ ^j„, kr^, 

 r 2 



and on dividing by k\ we may put the expression into the 

 form 



f . kr cos Z^\ 



L _ f (r) + cos^ «/( r) r^ cos^ S I sm -^ 



~^ " ^ r * 2 • I ~~ l:Tcos Z 



Again, on looking at the values of the other coefficients 



