190 Prof. Poweirs Abstract o/M, Caucliy's 



functions to be subject to a particular condition, viz. that we 

 may have a function ru' (g), such as to give 



n(g) = fl'oj'ig); (42.) 



in which case it will be found that the expression (40.) will 

 be reducible to 



8 = ID- (g + /2/). (43.) 



Now, from this form it follows that if g and t receive the 

 respective increments Ag and At, the value of « will remain 

 the same, if we have 



A§ = -flAt; (44.) 



that is, the displacement a will be the same for a molecule si- 

 tuated at the end of the time t, at the distance g, from the plane 

 (16.), and for a molecule situated at the end of the time t+ At 

 at the distance g+ Ag. 



The motion, then, of a molecule m is immediately trans- 

 mitted to other molecules situated on the side on which the 

 values of g are negative ; and the velocity with which the mo- 

 tion is propagated in the direction perpendicular to the plane 



(16.), which is expressed by the value of —-, given by equa- 



tion (19.), will be exactly equal to the positive constant /2. 



Again, it is evident, from the form of the functions (36.) 



(37.), that they have the same recurring values when we sup- 



27r 

 pose g to increase by -r— and consequently the function (43.) 



2 If 

 will do the same when g is thus increased, and t by -rrTr' 



Let us assume 



1=^ (45.), and 'T =~ (46.) 



If now, at the end of the time /, we divide the space into an 

 indefinite number of lamina? by parallel planes corresponding 

 to the values of § which reproduce the periodical equal values 



dti . . . 



of 8 or of -TT, then it will evidently represent the thickness 



of each lamina, while T represents the time of the isochro- 

 nous oscillations performed successively by a molecule. We 

 will call these laminae " plane waves," and we will suppose 

 their thicknesses divided into two equal parts by that one of 

 the parallel planes whose equation is 



ax-\-bi/-\-cz = g = —Sit, (47.) 



