370 GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



vt — v ?™ 



>?:„ = r£„ (28) 



>?.. — V 6m • 



Perform the same operations on the rf functions, and call the results 77" , 17^, 77". 



(29) 



This operation can be carried on indefinitely, and we shall have in each case of an 

 r) with an odd number of accents that the quantity R, or the expansion, as we may 

 call it, disappears, while for the tj's with an even number of accents it appears in the 

 form 



V dx 



k being here an odd number equal to the number of accents diminished by unit} 7 . 

 For an incompressible fluid we have therefore a set of very simple formulas giving 

 the relations between these various orders of motion. 



"We shall now introduce the equations of motion ; these are 



dU (hi x du, duj . duy 



dx\ ~ lit + "1 dx\ + " 2 7u, + h lt " dx n 



dU du. z du., du 2 du. 



= *■ + "1 *; + ««£ + •■• + ''» ,>,, ( 30 ) 



dx., dt ' 2 dx. 



dU du„ du n du„ . du n 



dx~ n ~~dt +Wl dx[ + n - dx~,^ + "" dx n ' 



Introducing the quantities £ and q and writing XT — q= W, these equations may 



be written 



dW du l \ t 



•^ = -ft + «2 £l2 + «3 &B + ' + M » &« 



7>>„ = "J — "1 £2 + ? '3&3 + r- w« &• (31) 



« . - • 



dx~ ~ ~dt~ «^fl n — «2f2n — — «»-l?»-l,n« 



