acted upon by the Undulations of an Elastic Fluid. 351 



character of these terms it is evident that the small terms that 

 are added by this operation will all be periodic, and have as much 

 positive as negative value*. The case will not, however, be the 

 same with respect to the equation (B) after a like substitution. 

 From what has just been said, the terms of the second order in 



the value of -^- will be wholly periodic ; but besides these there 



will be of the second order the terms 



2 + 2 \ dx* + dy* + dz* ) ' 

 Now, to the first approximation, 



d*a_ 2 /iV d?a rfV\ 

 dt*~ a ' W + dy* + dz*)' 



Hence, by the same reasoning as that applied to the function yfr, 

 we shall have for different axes of vibrations, 



a=a l + <r 2 + <r 3 + &c. 

 Consequently 



cr 2 = a j 2 + <7 2 2 + o" 3 2 + &c. + periodic terms, 



dy* dy dW dy 



£> = -aW + W + V + &C " + Pen ° dlC tCrm ^ 

 &c. = &c. 



Hence, by substitution in the terms of the second order expressed 

 above, it will be seen that the general value of <r will contain 

 terms that are always positive, and that these terms are the sums 

 of the terms that would be applicable to the several component 

 vibrations if they existed separately. So far, therefore, as regards 

 these condensations of the second order, the law of coexistence 

 holds good, just as for those of the first order, and as indepen- 

 dently of the relative positions of the axes of vibrations. This 

 result, it is true, rests on the supposition that udx-\-vdy + wdz is 

 an exact differential to terms of the second order for vibrations 

 about different axes ; but since this is an a priori analytical hypo- 

 thesis, the only inference to be drawn from this circumstance is, 

 that the law is general with respect to vibratory motion, and in- 

 dependent of arbitrary conditions. 



The law of the coexistence of vibrations having been thus as- 

 certained to terms of the second order, we are prepared to form 

 the equations applicable to given cases of motion in such manner 

 as to take into account the general composite character of the 

 motion of the fluid. With respect to this part of the investigation, 

 I have seen no reason either to add to or correct what is contained 

 in Parts It. and lll.,and shall therefore proceed to the third matter 



