378 GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



the quantity (64) is = A ik , therefore (65) becomes 

 From equations (58) we have, however, 



= -* + *i s; + ** ^ + • • • • + *»-. -^- 



Substitution of this in (66) gives a series of terms of which the first is 



This is = for all values of t other than i = 1. Similarly for all the terms until the 

 I th ; the coefficient of*; is then = A lh . Now this substitution gives before reduction 



o = 22 as [-4?, (*i ,/,; + ** ,/,; + ■••• + *»-i ^ + ^r)J (67) 



on performing the above described reduction this becomes 



2^(^*0 + 22i(i -Ji*) = o (68) 



Since the first and second minors of J, i.e. A lh and A'{ h , all satisfy identically the 

 equation of continuity, (68) may be written in the form 



t/4> 2 . rZ<K . . . (Z* 2 . 



Ai ^ + 4i 3^ + •••• + An s; + e 2 = o (G9) 



A n "^T + A " 2£~ + ■•••+ 4- "7/xT + ~ ° 

 in which 



e, = "|2^|^ (70) 



Again, by aid of (53) we can replace (69) by 



d^ d*! d$i Mb _ 



A u d Xl + ^12 cfc "+" + A i» <&■„ ^ A — A i 



<7* 2 >i ^5? . _i_ 1 d ® 2 I f?$2 — Y 



A v dx r "+" ^n </. r , i-- -- '-r-^ ln ^ -f ^ — A n _j 



