382 GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



venient from considerations of symmetry to write 0, 1} 2 • • • • n -i as our functions, 

 for the same reason it will now be more convenient to write U 2 , 3 . . . . 0„. In all 

 that precedes, then, the function is identical with the new X , and in general t _ r 

 (old notation) = t (new notation). The minors of J corresponding to 



d0 1 dOi dOi 



dx 1 dxt dx n 



are A n , A 12 . . . . A ln respectively; and similarly for the other rows. 

 If we solve the equations 



d9 l . d0 1 d9 1 d» x _ 



W + u, 7u t + «. 3£ + • . . • + «. s: - 



dft, . d& 3 dd a , d0 2 _ (8M 



we find 



d6. , d9 n d6„ d6 n __ 



■dJ + ^dJ l + ^d^ 2 + ---- + n "dJ l ,- 



J "i - ~ \ dJ A * + a ^ 21 + • • • • + a A \ 



JUn = — \ Tt A *« + ~di A * n+ ••••' j ~~dt ^'<»$ 



Substituting these values of u in 



^ii _l — 2 4- -4- — " = n 



dx, ~^ dx,^~ T r/.r„ 



and we find by the rule for differentiating a determinant 



dJ dJ , A/ _ <W /OKX 



•h s; + * 35 + • • • • + M « &. - ~ * (87) 



or J" satisfies the equation 



ZX7" 

 dt 



(88) 



2) 



It would be interesting to determine the effect of the operator j t upon the various 

 minors of J, but it is sufficient for the present purpose to find the effect produced 

 upon the first minors by this operator. The method of doing this is, in the case of 



