262. Ona Theorem relating to Equations of the Fifth Order. 
fication, that with the system of equations 
a=m+/O + Vp+qv@, &e., /OV p+qV OV p—qVvO=s, 
any function , 
f(a, B, ¥, 6) + (8, y, 9; a)+ oly, 6, a, B) + (6, a, B, Y) 
is a rational function. } 
The coefficients of the quintic equation for # must of course 
be of the form just mentioned; that is, they must be functions 
of a, B, y, 6, which remain unaltered by the cyclic substitution 
a8yd. To form the quintic equation, I write 
0—x=a, 
Aa? BF y5 8? = b, DS*a? BFy?= C, BR*y28? a? =d, Cry? 83 a5 8? =e; 
then we have 
O=a+b+c+d-+e, 
and the quintic equation is 
fl fo fo? fo® fo*=0, 
where is an imaginary fifth root of unity, and 
fo=a-+ bo + cw? + dw? + ew*. 
We have 
fo fot= 0? + (o +.04)>!ab + (@? + o°) Zac, 
fo? fo® = Ya? + (w* + w*)2ab + (w? + w*)2/a0, 
where &! is Mr. Harley’s cyclical symbol, viz. 
Dlab=ab+be+cd+de+ea; 
and so in other cases, the order of the cycle being always abcde. 
This gives 7 
fo fo*fo* fot = Lat + Ya?b? — Lab + 2Za*bhe— Labed 
—52z/a?(be+ cd) ; 
and multiplying by f1, = =a, and equating to zero, the result is 
found to be wh 
+a°— 5abede—5>'a3(be + cd) + 5>/a(b7e? + c7d*) =0. 
Or arranging in powers of a, this is 
od ee 7 
+a?. —5(be+cd) | 
+a’. 5 (bc? + ce? + ed? -+ db*) 
Ce ere 3 
ry : 5 (bc + c8e + e8d+d°b) 20: 
+5 (be? + c?d*?—becd) 
6+c+e+4d° 
ae { —5(b%de + c®bd + e?ch + dec) 
+5 bd2e? + cbh2d? + ec? b? + de*ec? 
