720 
Proceedings of Royal Society of Edinburgh. [subs. 
Ej_ = - tD 
From these it follows that 
E x + E 2 + . . . + E^_ x = E = — 2§ f . L) 
where S t = 0 or 1 according as t is odd or even. 
Similarly, 
C= -28 S _ S ,A 
where S s _ s , = 0 or 1 according as s — s is odd or even. 
Under the conditions of axisymmetry the left-hand sides of I. 
•and IT. vanish and we have 
A+B+C=B+D+E=0 
A+C=D+E 
B = ( - l)d) = ( - 1) S-S 'A 
and 
or 
( 1 ■■ ■ Z 
2r t + s | s | A 
\ -1 A 
a l a 2 / 
/2r\t’ + s\s \ 
f% r 1 1 + s \ s j A 
\ a l a 2 J 
\ a l a 2 V 
/2 r I F+~* 1 8 x 
\/2 r 1 1' + IIJa 
V a i a 2 / 
/V a ] V 
/2r a + # | s x 
\/2r 1 1' + s | A 
\ tt l a 2/ 
a i V 
fYr\t’ + s\( 
'2r\t' + s\s\d\ 
\ a i A 
0*1 0^2 ^ ) 
/2r 1 1' + s j s N 
\/2r | if + s|s|A 
\ «i a 2/ 
/\ a i a 2 i J 
III. 
It will be observed that the 2r numbers representing the rows 
and columns in the aggregates III. are divided into three groups 
" d -CvV ';)■ «* - »< 
these aggregates the group a is constant as rows on both sides of 
the equation, on the one side the group b is constant as columns, 
while the group c is divided up between the rows and columns, 
