(4944 9) 
OP, OP, (OE er 5 : 
ay 1 Ge = (Ge ay) 
OP). AAN CEA : 
v 1, —— TS ne ES TE. Dd vG . 
RT iy (5 u ol (4) 
(OE tin Oy DoD Es ie NC hie) Cone 
Oy pe on Oy ; 5 =) roa 
ete. 
4. Examining now the possibility of a series of homogeneous 
functions P; satisfying the conditions (4), we introduce the values 
X = aa* + 2b ay 4- cy’? Y = dae? + 26' ay — ay? 
Oa coy 
== la db) 2(b—a)y. 
Or OY 
Putting 
Pi =p @ + Py 
the first condition gives immediately 
p, =4(a —b) p, =A(a + 0). 
Thus the function P, divided by 4 
P,=(d — Ba + (a HD) = por + py 
always exists. Proceeding now to 
P,= qe + gay + gy? 
we find that this function exists when the coefficients satisfy the 
following equations 
q, = (8a + 2b') p, + ap, 
29, — 29, = (4b — 2a’) p, + 2 (a + WU) p, 
—g, = cp, + (2b — 3a’) p,. 
Therefore it is necessary that 
(8a 4 2b'+ ¢)p, + 2(b— a) p, = 9 
or that 
(a — b) (a + c)—0. 
By hypothesis a + ¢==0, thus the first condition is 
ge Be Oe ko ina NE 
If this condition is fulfilled g, may be chosen arbitrarily, for 
instance g, = 0, and we have 
P, = (a+ 0’) [bay + (a + 25) 77]. 
From this form it is evident that this function and all the following 
functions P, P,... vanish when 
Ra: BO.) ei a 
