248 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vm Analogous to the definition of trigonometric functions in terms of exponentials, he defined quasi cosines and sines by and Tqx as their quotient. Their relations are discussed. He defined pseudo cosines and sines by Cpx = Cq[(p -1)*] = e~*Cq%xy Spx = - e~xSq2x. For each prune p<100, he gave (pp. 65-86) the (integral) values of ex, ind x, Cqx, Sqx, Tqx, Cpx, Spx for z=l, 2,. . ., p-H. L. E. Dickson95 extended the results of Serret74 to the more general case in which the coefficients of the functions are polynomials in a given Galois imaginary (i. e., are in a Galois field of order pn). For the corresponding generalization of the results of Serret79 on irreducible congruences modulo p of degree a power of p, additional developments were necessary. To obtain the irreducible functions of degree p in the GF[pn] which are of the first class, we need the complete factorization, in the field, h(zpn-~z-v) =IL(hpzp-hz-/3) where hv is an integer and ft ranges over the roots of all of whose roots are in the field. For the case v = Q this factorization is due to Mathieu.73 Thus hpzp—hz—/3 is irreducible in the field if and only if BT^O. In particular, if /3 is an integer not divisible by py zp — z—/3 is irreducible in the GF[pn] if and only if n is not divisible by p. R. Le Vavasseur96 employed Galois imaginaries to express in brief notation the groups of isomorphisms of certain types of groups, for example, that of the abelian group G generated by n independent operators ai,. . ., an, each of period a prime p. If i is a root of an irreducible congruence of degree n modulo p, and if j = ai+ia2+ . . . +?)n"~1an, he defined aj to be aiai . . . ana». Then the operators of G are represented by the real and imaginary powers of a. A. Guldberg97 considered linear differential forms dkii dy with integral coefficients. The product of two such forms is defined by Boole's symbolic method to be "Bull. Amer. Math. Soc.,3 , 1896-7, 381-9. »6M6m. Ac. Sc. Toulouse, (9), 9, 1897, 247-256. 97Comptes Rendus Paris, 125, 1897, 489.