384 



GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



« a i - j x " ^ + Xi2 ^ 2 + — + Xi " ^;j rfSi 



rfft 



e»„ 



<£tj 



rfft, 



£?Ct2 — ) ^21 J " "T ^22 rf- r +•••• + ^2 „ . } CIS. 



(92) 





The values of the X's are known to be 



Ax 



A ., 

 X u = 2^', X 12 = ^, etc. 



where 



(93) 

 (94) 



2q = A\ l + A% i + . ... + A\ n . 

 Similar expressions hold for X^ (; = 2 , 3 . . . . n and i = 1,2 .... n). The quantity 

 2 q can be expressed by a determinant viz. 



' £; Q23 ■ • • • tr 



Cg2 ('33 . . . . C 3n 



2q = 



in which 



c n2 C «3 



(95) 



coo = 5 (^) , C33 = 2 (4) ' ete - 



_ ^, d0„ d0 3 _ -^ d0 2 dOt 



~~ 2< d^. d7, c u — 2< ^~ (j. etc. 



(96) 



^ efe, rfx, 



j taking all values from j = 1 to _;' = n ; and c» = c M . Substituting the values of 

 X n , X 12 , . . . ., X ln from equations (93) in the first of equations (92) and we find at 

 once 



(97) 



2q,dcc 1 d$ t 



ds 1 = —j-,-2q^; 



similar expressions are to be found for ds 2 , ds 3 , . . . . ds n . Denote the projections of 



c?Sj on the axis by 



dx<t\dx%\ <fej»; 



then from what precedes 



u-idx^ + i^tfa^' + .... + Undx^ = 



<#! 



rf0 2 d0„ dO, 



dx, f dxj ' • • •' dJ n 



dh d6± d0^ 



dx t ' dx a ' ' ' ' '' dx n 



(98) 



