1915-16.] On Systems of Partial Differential Equations, Etc. 309 
we find that the present solution can he thrown into the form (9), where 
<E> — ■ cos 
0 2 G/ a 2 G/ 0 2 g/ a 2 G/ ' 
Jdrjpdrjq d£pd$q d£pdi)q d£qdr]p_ 
' 8 2 G Z * 0 2 G Z * ' 
MpO^q + dr) p dr) q _ 
S 3 2 G a . 
sm fe 
* 0 2 G* 0 2 G * 0 2 GA‘ 
+ 
+ 
pdyq d£qdyp d£pd£q dypdyq. 
9 2 K* 0 2 K* 
Writing the last equation in the form $ = (G) + (K), where the first term 
depends only on G and the second only on K, we may regard the equation 
(G) = (K) as a partial differential equation to determine K when G is given 
and to determine G when K is given. li pfq the equation is soluble in 
both cases, hence the part of M depending on the vector G can also be 
expressed by means of a scalar of type K, and similarly the part depending 
on K can also be expressed by means of a vector of type G. If p = q the 
part depending on K vanishes altogether. It should be noticed that the 
same value of M is obtained by writing 
M rot 2 Lp=- -^p + grad^, 
c dt q 
L p = - ^ + i I0t P& ~ g rad^K, A p = — div p G + - 
c dt p c dtp 
The lack of symmetry in the expressions for Lp, A^ and L g , A q in terms 
of G and K is worthy of notice. 
The theorem enunciated at the beginning of this paragraph is a 
particular case of the following theorem, which may be regarded as a 
generalisation of a theorem given by Appell.* 
If a vector L and a scalar A satisfy the system of equations 
i rotpL = - — + grad p A, 
c dt p 
(p=l, 2 , n) . (10) 
t . 1 0A A 
div p L + - — = 0, 
c dt p 
then L and A are right-handed multiple wave- f 'unctions . f To obtain 
Appell’s case we must put n = 1 
A solution of the above equations may be obtained by writing 
■ 1 0G • . , n A . V ^1 0K 
L = - -^-r+i rot G 4- grad g K, A = - div 2 G - — , 
c ot q c dt* 
where the vector G and the scalar K are right-handed multiple wave- 
functions. If we adopt the expressions used previously for G and K and 
* Bulletin de la Societe mathematique de France , t. 19, p. 68. 
t This means that each component of L is a right-handed multiple wave-function. 
