16 
THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. 
or by expanding 
5 
3 
(15) 
which gives 
(16) 
By the use of symmetric function tables, noting that a K = 0 when K> 3, we 
may write 
= <4 12 — 2 (- 4a|a 2 — 2 a|a? + 4aia|a! — a 3 a$) 
+ 3 (— 5a|a! + 5aj$a| 4- 5a|a 2 a? — ba 3 a 2 a x + a 2 ) 
+ 4 (— 4a§a! — 2 a 3 a 2 2 + 4:a 3 a 2 a\ — a 3 a\) 
— 6 (— 3a| + 7a 3 a 2 a x — a 3 a\ + 2a 2 — 4 a\a\ 4- a 2 af) 
4- 9 (5 a 3 a 2 — 5 a 3 a\ — 5a 2 a x 4- 5 a 2 a\ — af) 
4" 8cl 3 4- 12a 2 + 18cii -1- 27 
and since = —3, a 2 = —2, and cr 3 =l, 
R Ml = 2619 
12 To read the value of Saja|a| from the fifteenthic, we use the numbers in the row designated 5 3 
for coefficients. The first coefficient is in column headed 15 and therefore signifies—5a 16 . Thus we have 
2aja|a|= —5a ]5 + 5a u a x + 5o, 3 a 2 + 5a 12 a 3 + etc. 
