Algebraic Solution. 38] 
which may be given to it when one of the roots is supposed to 
become fixed. 
The form here used will admit of an exceedingly simple repre- 
sentation if, throughout my “ Observations,’”’ we replace 0, by 
@,, and vice versd. This requires that in art. ‘48 (vol. xviil. p. 52) 
we write 
Oras =O's, Ofas) =O, 
a change of definition which I shall accordingly suppose to be 
made. 
Further, I shall suppose that we replace 0, by 0;, where i is 
an imaginary suffix defined by the congruence 
t+ a=z1 (mod. 5), 
a being an integer. Or, if we agree to regard the infinite suffix 
oo as satisfying the congruence 
o-+a=o (mod. 5), 
we may replace 6, by 9,,. Lastly, I shall suppose the suffixes, 
after these changes, to be taken to the modulus 5. 
The changes being made, it will be found that all the fune- 
tions y are deducible from the expression 
0; 82+ Oa41 9044+ 80429043 
by writing, successively, 0, 1, 2, 3,4 for a. In fact we have 
9;09+ 994+ 9,0;=y2=1(xo), 
9,0, + 9,9) + 8,0,=9,=7(2)); 
0; 05+ O30; + O,99=Yo=7 (Xo) 
9,0; + 0495+ 697, =Y3=r(#s), 
0,0, +0,03+ 9,0,=Y4=" (2,4) 5 
and if, in these equations, we change 
“ 
‘ 0, 9, 93 8% Yo % 
into 
Oe 95, Oy 5, 5 2s 
respectively, we shall be reconducted to the system of art. 62 of 
my “Observations.” More extensive changes in our funda- 
mental formule and definitions would enable us to express the 
system with a greater concinnity between the suffixes of 0 and 
those of a, and provided that, on the right of the last system, 
we interchange 2, and 2, the system may be deduced from the 
equation 
0:0.+ 60419044 + 904200+3=YVy=T (aa). 
If, in the expression 
2 0. 6, =f 644104 +3 =e 64420044) 
we make a equal to 0, 1, 2, 3, 4: successively, we find the follow- 
