128 THE ROYAL SOCIETY OF CANADA 
(C) ant «te 1) m0 
D + Bu, 0) 2 =0. 
But (x, y) are linearly independent See of (A) and consequently 
cannot also be solutions of (C). 
If the above values for — j =, tc., be substituted in (A) and 
(6), and cognizance be taken of the fact that (x, y, z) are solutions of 
(A) while (x, y) are also solutions of (7), we shall find 
xr + Qxunis + YAt = 0, 
(16) xixer + (aie + x291)5 + Piaf = = 
o ’ 
xor + 2xoy2s + yt = 0 
The first and third of these are equivalent to 
Vive 
(pS | a 
ce aI ae 
9 X1X2 ) 
AAC V2 
while from the second in conjunction with these we obtain 
oe == 
2oJ 
Therefore 
Pine Ray a 
(17) rt G - 
It remains to be proved that the product 
HOF 
is constant. In order to prove this it will be sufficient to show that 
ube ora 
Oia, TOURS 
Now 
oH 0 
ai = D (x Y2 — X2 Yu) $I, 
OE ais FAURE dg 
Sais D (X1 22 — X22 31) + J 30 
Since x and y are solutions of (A), these equations become 
& (& 
oH 0 0 
= = O92 | Xu + a SA TT ale sts aaa Ge ) 
ou 
= 2 (p?— 9?) x2ÿ2 = 0, 
( oko) (2 
0H 0 
iar = px1 a ae ee ) —py1 Le + —— .) 
= 2 (p?— 9?) aim = 0. 
