210 MATHEMATICS, ETC. 
a set of simultaneous circular permutations; and that every circular permutation 
of (say) p letters is equivalent to (p—1) successive transpositions. The following 
propositions are demonstrated by Cauchy: 
Prop L—Jf a function of given letters remains unchanged by every circular 
permutation of p letters (p>8), it will also remain unchanged by any circular 
permutation of three letters. 
Prop. Il.—If a function of given letters can only acquire two distinct values by- 
any permutations of its letters, it is changed by a single transposition, and, in 
general, it is or is not changed by a permutation, according as this permutation is. 
equivalent to an odd or an even number of successive transpositions. Hence in 
particular : 
Prop. IIl.—A function which has only two distinet values is not changed by | 
circular permutations of three or of five letters. 
Serret proceeds (Lesson 21st), to examine the nature of algebraic functions. 
A function of any quantities, a, 6,c,.... is algebraic when it can be, obtained by 
performing upon them any of the following operations any finite number of times 
(1) addition or subtraction; (2) multiplication; (8) division; (4) extraction of 
roots with prime indices. These operations, of course, include involution to in- 
tegral powers, and extraction of roots with indices not prime. A function inyolving. 
only the operations (1), (2), is a rational and integral function of the quantities ; 
involving (8) also, it is rational; involving all four, it is general. If, then, 
A, B, C,.... denote rational functions of a, 6, ¢,....; p, g,7,-+---, prime num- 
bers; f, the operation of forming any rational function: then 
WACOM CY Bete gercen Pade nA AA Orv Fn B)} 
is called a function of the Jirst order. 
If A,,B,, Cy, .... denote functions of the first order; s, é, .... primes, then 
Gi CRS an ORME SEINE MONS: Be 8) 
is a function of the second order. And, generally, a function of the uth order 
will be of the form 
f (h, kU, 1... 0-H, aK, ...) 
where f always denotes a rational function; H,K, .... are functions of the order 
p—l : p, gq, ---- are primes; h, k,/,.... are functions of the (u—1)th or lower 
orders. From this form any radical, which can be expressed rationally in terms of 
the other radicals and quantities, can be eliminated; and ultimately it is shown 
that a function of the uth order can be thrown into the form 
1 2 n-1 
a + pe + Bpn + cagoeac el be gre 
where a, 8,....A are functions of the order; ”,a prime; p,a function of 
order (u—1) whose nth root cannot be expressed rationally in terms of a,8....A.* 
* Serret makes a further distinction among functions of the same order as being of differ- 
ent degrees, but his definition is strangely obscure, and this distinction does not appear to 
have any effect whatever on the subsequent reasoning. His use of the term degree is also. 
inconsistent with the sense in which the word is employed in Wantzel’s memoir.. 
