Prof. Cliallis on Hydrodynamics. 41 



pound velocity, and cr l the compound condensation, by the theo- 

 rem just cited Y 1 --=/cao- 1 . It is here, however, to be remarked 

 that since V, is wholly periodic, having as much positive as ne- 

 gative value, the same must be the case, according to this equa- 

 lity, with respect to the values of cr v These values of cr, are 

 therefore defective, not embracing the terms of 2 . a which do 

 not change sign. At the same time they include all those terms 

 which have the same constant ratio to corresponding terms of 

 2 . w, as, for instance, the first and second terms in the series 

 for a 1 . For the sake of distinctness, let us leave out of conside- 

 ration terms involving higher powers of m than the second, the 

 problem which is the ultimate object of these researches not re- 

 quiring such to be taken into account, and let us suppose that 

 a l -\-A<r l is the complete value of the compound condensation, 

 Ao-j being proportional to the sum of the terms of 5) . a which are 



2-7T 



of the form Am 2 sin 2 — (z—/cat-\-c). We shall then have 



A 



Vj = Kaa l =f(z — icat + c,) . 

 Similarly, since different eompound motions may coexist, 



V 2 = — Kacr 9 =Y(z-\-Kat-\-c 2 ). 



the motion being impressed so that the propagation is in the 

 negative direction. Hence if V=V 1 + V 2 , and o- = o-j + o- 2 (the 

 letter <r from this time receiving a new signification), we obtain 

 from the above equations 



dz dt 



This is the equation (7) obtained in Part II. of the former Re- 

 searches. It takes account of the composite character of the 

 motion, and to the first order of approximation is of general ap- 

 plication, as I have indicated in Part II. ; but it does not include 

 the quantity Acrj. This part of the condensation, which, as I 

 have before remarked, has the effect of making the excursions 

 of the condensed particles equal to those of the rarefied particles, 

 may be taken account of in a subsequent stage of the argument. 

 It is in respect to the composition of this quantity that the pre- 

 sent investigation is a modification of that which is given in 

 Parts I. and II. In the October Number I have made use of 



the equation 



ys 



Ka 



S and V representing the composite condensation and velocity, 

 and have supposed that A<7j is proportional to the second term 

 of this equation. But the present argument has shown that it 



