550 J. D. HAMILTON DICKSON ON RELATIONS BETWEEN FUNCTIONS 
whence, 
1s u u “ms 
oe om ee S a Sp2 =a tig teas teas tio, ty-2 
k-1 5-2] | ¢ t. t, t” eo t” R-1°k=2 
k—-29 Cx—2 k-29 Ux—2),| Yeo » Y—2 , ” i 
: é Sz_25 Spay Spo» Sx_2 
Sp_o) Sp—al: |Se-2's Spa} |Srne » Spao 
/ 
, ) ty-9 » bios ti» 
5) a) ’ Sr-2 ’ 
f u 
» Sp-2 Seo» Seg 
ti-2 5) a 
7 Ww 
Sr-2 5 Sz-2| [Sr—2 » Sk—2 
and, finally, on replacing the ?’s by the s’s and reducing, we have 
“ 
1 By i! ail © gil 8G he) Se 2a 
Spa | Se -3 9 Se » Ses » Si-3 » Sz-3 
Sp—1 Sp—2 s. < ee so" 
Sz-2 9 Sz—2 » Se-2 » Sz-3 > Sz-2 
, w mw 
» Sz—3 9 Sz—a > Sz-3 > Sx_3 
/ mn, ” 
» Sk-2 5 Se—2 » Se—2 » Se—2 
yd u” 
> > Sz_2 » Sz_2 5 Sk—2 
The process of reduction is general, and may be continued to any extent. 
As an example, and to return to the third set of equations for the sextic. 
$= 1 a BD. yi HOG 9). LOG, a ? 
&. 553/15 5e|,|6,10d|,|b, 10e|,|b, |: 
piasoll) a,10c| |a@,10d| \a, 4 
a 5b ~~ £0E Gand 
5. | bay Mae bi 08) o\ibg. 108 |, |b 5 BF 
a, 5b a,10c|>|a@,10d|>\|a, 5e 
: : 5 WO 6, 10d|,|6,10e 
a, Sot ta, 106} a, 0d 
which reduces to 
sje | bE Des 1070), 102, By Gz : (37). 
a? , 8335 
a,6b,10¢ ,10d, be, f 
vp Oy Ge 40d; 10, bf 
fy (2, Bb, 10¢ , 100, De 
es beglte $56 , LOO a0 
phe © 4 opp , We pig 
