184 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
sequently, since 7 and 7’ are complementary to each other 
(for their sum=>2,7,=6, a one-valued function), it fol- 
lows that rr’, and therefore 
a —5@64+582+ 57’, 
or 74(#), is, as we have otherwise proved in the last article, 
a six-valued function. 
17. The six values of rr’ may be exhibited in two ways. 
First, 
(@y Hy + Ly Xz ty Xp + a4 X3 +25 X,) + (Ly %yt+%y Xt %yX3+ LyX, +Xs Xo), 
(2 Ly + Hy Ly + Ly X3 + Us Ly t+ Xj) + (Lj Ay t+XyUst+ My Xy~+XLz%+AyX), 
(ay y+ Wy Uy + Hy Wy + Hy Xs + Ws Xj) + (# Hy + LyX; + @yL\ + X3%y+H;X%), 
(ay y+ Vy My + Hy U5 + Xs U3 t+ X3 HX) (@ Ly t @y Ly + UyXy +L %+Lz Xe), 
(ay Ly + Wy H+ Hl; y+ Ay Ly t ay Xy) + (¥, Vy + LyX + M5 Xj +X Le+Xy 2s), 
(ay yt ay X5+ Hs y+ Hy Ly + Ay X,) + (Up + Ly My + U3 Ly + Hy Lz+Xz 2s), 
corresponding respectively to the cycles 
XY a LX, 
Sees en Psy rebels LM peters 
ee mee era as ae aM 
, XN ¢ \ 
‘ / ‘ / ‘ 
X,.f Xo Ly ‘. 1 bey 
oy eae: 2 % 1 \% 
H 1 I \ 1 1 
1 1 ! 1 1 
\ ! ‘ i \ ! 
\ / \ ' \ ui 
‘ ’ \ / \ i 
N. ¢ AY 7 \ 
BS “, ‘s Aen XS ra 
preteen BL 
ee. % we 
Pa seas Ber ta Oe 
ie a f Sat as. y yer se. 
2z,/ Ne A yg be ee \2. 
3] \ tA 41 y os aT eat 
! 1 ' ! 1 
H 1 1 H I ' 
1 ‘ 1 H \ ‘ 
\ i \ ‘ i 
7 ‘ i ‘ a 
\ 7 ‘ a \ 7 
‘. 7 Lg SS ‘. Pd 
See aw oe 9 ™ —— | ae pet - 
S 4 pe ea X, orn 
or, secondly, 
(@y Lat Ly Uy t Ls V+ Le X3t Uy As) + (@, Le +43 Xz + %5 Lat Ly M+ UN), 
(@y y+ 24 X54 Uy Hy +Ly%yt a5 Ls) + (@ Ly Uy y+ Ly @y+ %yUzt+Xs%), 
(2 y+ 23 Ls + Wy Vy + Hy + Hy Hy) © (1 yt Hy Hy + Ly Xs +5 Uy+%, 2), 
(@y Ly + ay @5 4 Lz Ly + ay Lys as) + (LT, Lp +a, Xt 3 %y4+LyXs+H5X), 
(ay y+ Lg V+ Hy Ly + 4&2 +L5 Ls) + (ay 3+ a5 Xy+ Ly yt LL; +52) 
(2, y+ 4 Ag+ Hq H+ Hy H+ X54) + (X, Xp + ay H+ XyX3t+ XyUz+H5 2), 
