a ii 
TRANSACTIONS OF SECTION A. 413 
And, writing 6 = 28 .«,, d = 2° . w,, then 
& = w, with B = 5 gives n’ 
& = , with B + 5 gives ¢/ 
& = w gives one of £’, n’ = 
3 (& + 1) usually. 
4(é + 1) usually. 
(— + 1) always. 
ou Ha 
vi. The complete series of terms 1’, vu’, 7, v divisible by p is given by 
Rig) ote mé = 9, vey mé = 0, T 4 me = 9, VE + me = 9, (mod p) 
for all integral values of 7m, so that & is the period of recurrence. 
Thus ¢ may be called the Haupt-suffix in v, = 0, just as fis called the Haupt- 
exponent in 2* — 1 = 0 (mod p). 
2. The Algebraic Functions derived from the Permutations of any 
Assemblage of Objects. By Major P. A. MacManoy, F.R.S. 
3. A Mode of Composition of Positive Quadratic Forms. 
By Professor E. H. Moore. 
The n-ary quadratic form : 
n 
A= & a;;%;,%; (a,j =4;;), 
ij=1 
with real coefficients a,; is positive if, for real values of the » variables x,, it takes on 
only positive or zero values. From two n-ary quadratic forms A and B we obtain by 
multiplication of corresponding coefficients a third form C, the inner composite Ax B 
of the two forms A and B. 
For this inner composition of n-ary quadratic forms the property of positiveness 
is invariant, 7.e., the inner composite of two positive forms A and B is likewise positive ; 
otherwise expressed, the class of positive n-ary quadratic forms is closed under the 
process of inner composition of forms. For n=1 this theorem is that of the closure 
under multiplication of the class of non-negative real numbers. Necessary and 
sufficient conditions in terms of the matrix (a;;) of coefficients for the positiveness 
of the form A are well known, and for n=2 the theorem is readily proved by con- 
sideration of these conditions. The theorem in its generality is readily proved by 
consideration (not of these conditions but) of the fact that a form is positive if and 
only if it is expressible as the sum of squares of a finite number of linear forms with 
real coefficients. 
Our theorem: If A and B are positive forms, then A x B is a positive form, has as 
corollary and generalisation the theorem: If A”, A’ — A”, B’, B’ — B” are positive 
forms, then A’ x B’ — A” x B’ is a positive form. 
Similar theorems hold as to positive hermitian forms. 
These theorems have analogues and applications in the theory of linear integral 
equations. 
4. Proof of a General Theorem relating to Orders of Coincidence. 
By Professor J. C. Freups. 
Let f(z, vu) =u" + fp_yu" 1+. ..+/,=0 be an equation in which the co- 
efficients f, are rational functions of z. Suppose (7)’c°) and (7)(co) to be partial bases 
corresponding to the value z = ©°, the orders of coincidence furnished by these partial 
bases being complementary adjoint to the order 2. The general rational function 
of (z, u) conditioned by the partial basis (r)‘°°) we shall designate by the notation 
1 n 
R(- — Spy teen math ale. 
z,u , 
f= TF 
Tn order that a rational function wv (z, ~) should be conditioned by the partial basis 
