124 THE ROYAL SOCIETY OF CANADA 
~ 
(4) + gre — 2a’q3 = — + 719: —2an, 
0 
= + qits — 2b'q3 = a = r193—2bf2 . 
If the values for qi aid 1; oD a in terms of the function ¢ be 
substituted and cognizance be taken of the relation (1), we find that 
the condition equations are indeed identities. Therefore all deriva- 
: : : 4 0°70 06 
tives of higher order can be expressed uniquely in terms of AEE 
00 
D Furthermore it can be shown that the Normal System has pre- 
cisely three linearly independent solutions, apart, of course, from a 
constant which is obviously a solution. 
On substituting the values of g; and r, (i=1, 2, 3) in terms of the 
function ¢ in equations (2) and making use of equations (A), we obtain 
the following equations: 
3 2 
(5) oto log  . 08 9 9 nn, 0°60 
u*Ov 





Ou 07 ou dv dv ou du 0v 
0° ie 
0°60 paar 9 26. 00 dlog¢, 9 
dudv Oudv Ov Ou Ou ov Ou Ov 
GREEN 
ne 
Whence (on taking account of (1) and (A)): 
M0 0 DAop D 2070220 
Ou29v ou dudv 
0°0 CHOEUR 




Oudv dv dur 
Pup 
on (aaa) @ + 55 (Grae) 
du \ dudv av \ audv bi = 28 ee eae ote? av) 

aa 
and therefore by integrating this equation 
0°60 Cy 
(6) oudv 2 
where c; is a constant. 
It is evident from the previous discussion that two of the three 
fundamental integrals of the Normal System will be solutions of the 
equation (6) when c,=0, 7.e., solutions of the equation 
(lye — =0 
The general integral of (6) is 
(8) en HAE EN ge 

