82 
Proceedings of Royal Society of Edinburgh. [sess. 
These quantities satisfy the double series of equations 
Aa + B^3 + . . . 
. = K 
Aa' + B/T + . . . 
. = 0 
A a + B f3 + . . . 
. = 0 
Aid + B '/3' + . . . 
. = K 
A a. + A a + . . . 
. == K 
A/3 + A p + . . . 
. = 0 
Ba "4“ B ct + . . . 
. = 0 
B p + B'jB' + . . . 
. = K 
the second side of each equation being 0, except for the r th 
equation of the r th set of equations in the systems. 
Let X , g , . . represent the r th , (r + l) th , . . . terms of the 
series a L , M , . . . . the corresponding terms of 
the series A,B, , where r is any number less than n , 
and consider the determinant 
A L 
A (r-1) , . . . , 
which may he expressed as a determinant of the n th order, in 
the form 
A , . . . 
• , L 
o 
o 
AP, . . 
L (r~ 
o 
o 
0 , . . 
. . , 0 
,1,0,... 
0 , 
. . , 0 
,0,1,... 
Multiplying this by the two sides of the equation 
K 
a , /3 , 
d , ft , 
