416 SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
whence by equation (37.)) 
= — ^|o.4^.+ I .4 ^_i + 6.4^_2+17 .4^-31 
0 . 12^+3. 1 2^_i + 18 . 12^_2+51 . 12^_34-0 . 12^_4-f-3 , 12^_5 4- 18. 12^_6 4- 
-f-51 .12^_.7-1-0. i2^_8+3. 12^_9+ 18.124 ,_,o+31 . ] 24 ,_ii|. 
Lastly, for Z we have 
1 
144 
=\ — 0.6^— 1 — 4.6^_2+3.6^_3— 4.6^_4— 1.6^_5}-.Y2 
consequently 
Q^-, = i — 1.6^— 0.6^_i- 1 .6^_2— 4.6^_3+3.6^_4— 4.6^_5}..Y2, 
m=6, s = 4:, v=2, t=3, n=st=l2 
-n -4-A -_± 
%o — ]^f)5 %i — %2 j2’ 12’ ' 12’ — 12’ 
and therefore, by equation (46.), which gives the value of Z in this case, putting Z' for 
the first line of Z, Z" for the second and Z'" for the rest, 
Z'=mi-1.2.-0.2,_,-1.2,_2-4.2._3+3.2,_4-4.2,_3 
T44[ 
= ii4{(-'-*+3)-2-+(0-4-4).2,_, 
=— la -8 2 1=^--?^ 2 , 
Z"=(2,-8.2,_,).i|4[l2.12,+ 11.12,_,+ l0.12,_,+ 1.12._,4, 
which, putting for 2^ and 2^_i their values 
12,+ 12,_2+12,_4 + 12,_,o, 
and 
becomes 
12,_, + 12,_3+....12,_, 
Z"=Y^^ 12. 12,-88. 12,_, + 10. 12,_2-72.12,_3+8.12,_4-56.12,_3+6.12,_9- 
-40.12,_7+4.12,_3-24.12,_9+2.12,_,o-8.12,_„ ; 
and lastly, 
Z"'=Y^^l . 12,+0. 12,_, + 1 .12,_2+4.12,_ 
I r. 
+ j2[1.12,+4.12,_,-3.12,_2 + 4.12,_3+1.12,_4+0.12,_3+1.12,_6 + 4.12,_; 
+4[-3.I2,+4.12.^,+ 1.!2,_,+0.12._.+ 1.12,.,+4.12,_,-3.12,_,+4.I2,.; 
+ 1.12„,+0.12._.+ 1.12._„+4.12.. 
