14 



PROCEEDINGS OF THE AMERICAN ACADEMY 

 k n n— k (m j.^ ? Qn— fc-y 



i=0 p=0 f'r 



p + q = n—h, 



q dxvdy* ' 

 we may, from (14), calculate the b's in terms of the a's. 



Formulas for the coefficients of M(v) in terms of the coefficients of L(u). 

 p + q = n: b pq = (— l) n a pg . 



p + q = n — 1 : 



dcfp+i.g . da 



\<i = (- l) n 



[< 



dx 



+ 



p, g+i 

 dy 



J - a pq \ 



p + q = n — 2: 

 b m =(- 1)^ 



- 1) f^a p+2 , 



2! 



3 2 a p+li9+ i 3 



g + 2 - r^'*^ + 



-(*-!)( 



3a; 2 da-di/ 



dx 



! «p,<H-2 \ 



K15) 



+ I iTJ + ^] 



p + a =n — k; 



&™=(-D»22(- 1 )t 



(n - ! 



6 fc ? a p +i >g +jt— z— i 



'pa 



i=o i=0 



(?i— &) ! i ! (k—l—i) ! Ox^y*- 1 -* 



Assuming for the moment the fact, which will be proved presently, 

 that L(u) is the adjoint of M (v), we may obtain symmetrical formulas 

 connecting the a's and b's. For the formulas expressing the a's in 

 terms of the b's may be written down from those just given by simply 

 interchanging the letters a and b throughout. If now, from these two 

 sets of formulas we replace, in the identity 



(- 1)" a PQ + (- l) k b pq = (- 1)* b pq + (- 1)" a pq , p + q = n-k, 



on the left side a pq , on the right b pq , by their values in terms of the 

 b's, a's respectively, we obtain the desired symmetrical formula, 



k— 1 k—l 



2 2 (- 1)' 



1=0 i=0 



(n-QI 



d*-' b p+i , q +k—i- 



-f (- 1)* 2b 



pq 



(n — k)lil.(k — l — i)l dx*d y k ~i-i 

 = (— l) n +* times the same function of the a's and their derivatives, 



p + q = n — k.3 (16) 



3 It should be pointed out that these formulas are not precisely analogous 

 to those obtained for the second order. For, if we put here n = 2, fc = 2, we 

 get, using the notation employed for the second order, 



