EEY. E. HAELET ON THE METHOD OE STMMETEIC EEODHCTS. 
That is, the four horizontal rows read downwards are identical in value and order with 
the four vertical columns read from left to right, while Ic^ and Tc^ lie in inverse symmetry 
upon, and and around, diagonals. Probably no other distribution would render 
more nearly symmetrical*. 
Combining, as in former cases, the conditions of symmetry and rejecting incongruous 
results, we arrive at the quartic 
of which the roots are the unreal fifth roots of unity. Let then ^y, and co^ denote 
these roots, and let 
f {u)=x^-\-u ; 
then 
and 
f {(J^) =x,-\- 01^X2 -h X2 + + a'x^, 
= {%xy — b{^xf%X]X2 -\-6(%XiX2y + firr' 
where 
and 
(5“*— 5aJ^<?+5<zV)+5rr', 
fjJ — /Y» /y* /y» /y» _L /v* /y» /yt /y» -J_ /y* /y> 
• ■ 1 1 tv 1^3 j tv 2*^4 1 tv3tv5 I tv 4 wj tv3tv2* 
These two functions, r and r', are circular, and complementary to each other. Since 
then’ product is unsymmetric relatively to x^ it follows that '!rlx) cannot in general be 
rendered symmetric. Before proceeding to discuss this product, it will be convenient to 
introduce some other matters connected with the general theory. 
Sectio:n' II . — Circular Functions and the New Cyclical Symbol. 
7. In the transformation and general treatment of the higher equations circular func- 
tions occupy a conspicuous place, and play an important part. An attentive considera- 
tion of the structure of such functions will enable us to devise a calculus whereby opera- 
tions upon them will be materially abridged. The theory is far from being complete, 
and its practical application admits of great improvement. In my original memoir I 
have proposed and applied a symbol which not only helps, I think, to throw some light 
on the general theory, but also enables us to effect with ease and rapidity calculations 
which would otherwise be very laborious, if not wholly impracticable. The method 
there employed exhibits to the eye and to the mind the various combinations of dimen- 
* "WTien all the divisors of n are even, a cyclical arrangement must be adopted, as in the case for quartics 
(art. 4). When n is prime or odd, an arrangement similar to the above, or a modification of it, will probably 
be found available. 
