38G 



GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



dP 



d9 r 



dO. 



«i = ;&: + to tit. + to </:•; + 



t&'j 



dx l 



dx x 



' / + *i ^ 





(106) 



M» : 



tfP dS, <# 2 



^~ h to dF+fo£:_ 



+ to s: 



cte„ ' ri £&r„ ' Y ' 2 dx n 



and from these follows 



Ul dxy + Maefa-a + .... + u n dx n = dP + ^dd, + fal0 a + .... + $ n d0 n (107) 



This result may be obtained in another manner which will bring into evidence some 

 interesting properties of the functions with which we are dealing. The method to be 

 employed I shall merely indicate, as it is similar to that employed in obtaining 

 equations (61) and (62). The original equations of motion are 



now 



and consequently 



Du-i _ dU 



Du 



dU 



-jf ' — izr etc, 



dt 



dx. 



d _d6 y d dOz d_ 



dXi ~ dx t db] ' dXi dd 2 



Du 1 _de i dUd6 3 dU 

 ~dt ~ dx 1 d9 l ~> dx t d$ 2 ' 



Du, _d6 t dU dt^dU 

 W ' ~ dx 2 ~dT x ' dx 2 dd 2 ■" 



~* dx t d$ n 



+ 



d0„ dU 

 dx~ t d$ n 



.d^dU 



■ ~*~ dx, d6 n 



(108) 

 (109) 



(110) 



Du, _ d0, dU dd 2 dU dO n 



~dt ~ dx n d$ l "*" dx n d0 i ~*~ ■ ■ ■ • 



de„ du 



dx' d0 n 



Multiply these by A n , A 12 , . . . ., A ln respectively; then by A 21) A&, . . . ., A 2n re- 

 spectively, etc. ; add in each case and we shall obtain 



(111) 



