460 
ON THE ADDITION OF THE DOUBLE S--FUNCTIONS. 
[703 
then multiplying by 
we have 
-d 56. -S56 • 6^, — Va 5 b 5 c 5 a 6 b fi c 6 , 
DEtfi. j4 56 . -S 56 . C 56 = -pr~ {a 6 b 6 c 6 P a 5 b 5 c 5 ^Q], 
= —— {a — q.b — q.c — q.VP — a—p.b —p . c — p . Q} y 
or recollecting that e VP, e Q are = ap 3 + ftp 1 + yp + 8 and aq 3 + ftq 1 + <y ( l 4- 8 respectively, 
this is 
€. DEufi. -d.56. -Z?56. C'56 
= {a — q .b — q . c — q . (ap 3 + ftp 2 + r yp + b) — a—p.b — p.c—p. (aq 3 + ftq 2 + <yq + 8)}. 
Using the well-known identity 
ap 3 + ftp 2 + >yp + & = aa 3 + ft a 1 + 7 a + 8. C ——— 
+ g 63 + /№H-y» + 8. C -f-5T a --P 
c — b.d — b.a—b 
+ + p* + 7 o + 8. <l-p.a-p.b-p 
+ <xd’ + pd< + 7 d + S. — 
a — d .b — d .c - d 
and the like expression for 015 3 + /S5 2 + 7^ + 8, there will be on the right-hand side 
terms involving 
aa 3 + fta 3 + 7a + 8, ab 3 + ftb 2 + 76 + 8, ac 3 + ftc 1 + 7c + 8 : 
but the term in ad 3 4- ftd 2 + 7^ + 8 will disappear of itself. 
The term in aa 3 + fta 2 + 7a + 8 is 
t — j -o-q.c —q.b—p.c-p.(a — q.d — p —a-p.d-q), 
p — qb—a.c—a.d — a 2 2 ^ ^ r r 2/ ’ 
where the expression in () is =d — a.p — q: hence the term is 
aa 3 + fta 2 + 7a 4- 8 7 7 
= b-a.c-a .b-q.c-q.b-p.e-p, 
which is 
_ aa 3 + fta 2 + 7a -I- 8 D 2 ^ 2 
— T -^56 • ^56 • 
6 — a. c — a 
Forming the two other like terms, the equation is 
„ nr a z? n _ "h ^a“ + 7® "f 8 p 2 
e • • ^-56 • -¿*56 • ^5« — r ■ -¿*56 • t^ 56 
0 — a. c — a 
ab 3 + ftb* + yb + 8 r 2 2 
+ c-b.a-b 66 * 56 
ac 3 + ffc 2 + 7c + 8 2 ^ 2 
a — c.b—c 56 ' 56 '