376 GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



d dq d dq d_ dq 



- *i - s; jf. + s, ~pm + • • • • + dT„ jm_ • (55) 



dx x dx 2 dx n 



If we multiply equations (50) by A,_ 1A , -4g_i f a» • • • •> -4*_i,n respectively and add, 

 we shall obviously have 



*i = J- Mi-i.itfi + ^i-i.2 A" 2 + .... + ii-,,,/!,,!- (56) 



a form of $j from which some interesting results may be obtained. Substituting % : 

 for its value in (28) gives 



d(U- q) ='f ^ dxj + 1~ W (57) 



j = i i = i 



and from this we obtain at once 



<nu-q) dA a dO, d0 2 dO.-r 



—foT~ = ~dT + *i dJ, + ^ &i + • • • • + *»-! "ST 



d(u-g) _dA n de, d& s de n . x ._„. 



—st" --&- + ^s: + *as: + .... + *„_i -^- (58) 



d(U-g) dA x d$ l d0 o d$ n _ x 



dx a =-¥+ $ '^ + ^^ + --- + $ .-iX 



For the case of steady motion the first terms on the right side of (58) drop out and 

 the equations take the form 



d(U-q) d,6. . d$„ , d9„_! ,_.. 



Multiplying by (/a*!, c7.r 2 , .... <7x„ respectively and adding, we see that for steady 

 motion the functions <& are the differential coefficients with respect to 1} 6 2 , . . . ., 0„_i 



of a function. 



W= U-q= W{6A-- • .0»-i) 

 i.e. 



The operator 



dW dW dW ,„„, 



*i = -dg-, ** = W t - ••■*»-! = d5Z* W 



A — + A — + +A — 



An dx^ A ™ dx,^ ^ ^ ln dr. 



is well known to be equivalent to 



J-< 



J de 



