PARTICULARLY THOSE OF TWO VARIABLES. 
833 
w+w +w=w+w wv 
w+w+w= w+V V + w 
W+V V + W= W+V V + V V 
W+^* V + W= W+V V + V V 
W+«uV+W=W+« ! i V +0 B *v 
W+W +vV=W+0gjfc* +W 
<W+0. S V + W+^ 7 2 V + V V 
0fV+ V V + w=0A 2 +0 5 2 V +0/ V 
W+W + W=W+V +W 
vv+w+vv=w-w+w 
Then by (23) we have, for instance, 
4% s ® 13 ®' 12 = W+ W+W+W+0 3 2 V + w+W+W 
Ml” 0+ftr 0„ 2 ft'j'' ^I3 2 fti3 3 0/+ 2 0g 2 ftfi 2 ^lo’ftio 2 0] l 2 ft |', 2 
= 4(0 13 2 .3 12 2 + 1 W ^13 2 ^ 13 S ) 
by equations (205) and therefore 
c i3 2 ®is® , ia— 0ia 2 ^is 2 +0n 2 ^n 2 0 1O 2 V ^ 13 -ft 
13 
which reduces to an identity when f, rj are both zero. Another similar set is 
0„ 2 V +0/ V + V V + V V =0 5 2 V + v V + 6 *» +0 4 * v ~ 
0 9 2 V +0/ 3/ +ft, 2 v+W=*V fto 2 +03* V +0 U 2 V +0u 2 V 
0 8 2 V+MbZ+V V+0n 2 V=W +W+W +W 
08 2 V + W+01 2 V +06 2 V = W +0 1O 2 V+04 2 V +03 2 V 
0o 2 S5 2 +W +0 3 2 V + W=V V +0 8 2 V+V •*»* + w 
0« 2 fts 2 +W+07 2 V + W=03 2 V +0 a s V+0* 2 V +VV 
V V +W+V V + w = 0 / 3»* + W+V V +0/ V 
W +0/ V +W +06 2 V =V V+V V +0 8 2 V+0» 2 V 
W +0 6 2 V +0 6 2 V +0 15 2 V=0 9 2 V+V V +0» 2 V + W 
<W +0 3 2 V +W +0 3 2 V =0 9 2 V+V V +0/ V+0 7 2 W 
W+0.* V +0i 2 V +0n 2 V= W + 0 7 2 4 3 +0/ V +0 U *V 
0o 2 V +0i5 2 ^io 2 +0i 2 V +W=V V +W+V V +W 
(206) 
and by means of these we obtain, among others, the result 
Mhs®'^ 0 8 S V- 013#+0n 2 V-0A 1 2 . 
which as in the previous case reduces to an identity when f, -q are both zero. A set 
similar to the last, and likewise necessary, is 
5 o 2 
