acted upon by the Undulations of an Elastic Fluid. 349 



it follows, if Qj be a function similar in form and composition 



to Q, and if regard be had to the foregoing values of w } a>, and 



<r, that 



, ^ , 2 dQ, , _. 2 dQ , -, 2 dQ Y 



w' = 2i.w-\-7n z —r-i ay—A .co + m' —r> a' = 2, . cr + m'—r 1 . 

 dz dr dz 



From these results we may infer that, on proceeding to terms 

 of the second order with respect to m, the composite velocities 

 and condensations are not found to be equal to the sums of sim- 

 ple velocities and condensations, but to differ from such sums by 

 small quantities of the second order involving the functions Q 

 and Q,. Respecting these functions, it may be observed that they 

 are wholly periodic, having as much positive as negative value. 



But it is chiefly important to remark that the part of cr' which 

 is expressed by 2 . a contains terms that do not change sign, viz. 



and that the different terms which the symbol 2 embraces co- 

 exist independently of each other. A distinction, however, 

 should be made with respect to terms which have the same 

 value of X, but different values of c. For all such terms the 



If 



values of /and -4- are the same; so that 



J gdr ' 



(2 . [/sin o?]) 2 =/ 2 . (2 . [sin q{z-a<t+ c)]f, 



But by a known trigonometrical theorem, 2 . cos 9 r (^"~ ffl '' + c ) 

 may take the form 



(n + 2S . [cosq^-c^l^qiz-a't + e), 



n being the number of the terms, c l and c 2 any two values of c, 

 and 6 a function of the different values of c. Now, since the 

 constant m may be supposed to be very small, and there is no 

 limitation of the number n, and since also the values of c for 

 simple vibrations do not admit of determination, it follows that 

 there are no conditions by which the quantity X . [cosg^Cj— c 2 )] 

 may be determined to be either positive or negative, and we may 

 therefore suppose it to vanish. Hence if in the foregoing ex- 

 pressions for w\ co 1 1 and a 3 , f be taken to represent z — a't + 9, 

 and if we introduce within the brackets [ ] the factor v&, those 

 expressions will be adapted to the case in which the terms con- 

 taining the same value of X are grouped together. Consequently 



