ON THE THEORY OF NUMBERS. 511 
or inversely in the composition of F and ¢ respectively, f, will be taken directly 
or inversely in the composition of F according as Ox w; is positive or nega- 
tive. 
By this theorem, the problem of finding a form compounded of any number of 
given forms is reduced to the problem of finding a form compounded of two 
given forms. For iff, f,..f, be the given forms, we may compound the first 
with the second, the resulting form with the third, and so on until we have 
gone through all the forms, when the form finally obtained will be compounded 
of the given forms, as will immediately appear from successive applications of 
the preceding theorem. We also see that we may compound the forms in 
any order that we please, or we may divide them into sets in any way we 
please, and compounding first the forms of each set, afterwards compound the 
resulting forms. If any of the given forms are to be taken inversely, we may 
substitute for them their opposites (Art. 92) taken directly. We may thus, 
without any loss of generality, and with some gain in point of simplicity, 
avoid the consideration of inverse composition altogether ; and, for the future, 
when we speak of the form compounded of given forms, or the class com= 
pounded of given classes, we shall understand the form or class compounded 
directly of the given forms or classes. 
110. The solution of the problem of composition given in Art. 108 may be 
put into a form better suited to actual computation. 
The system (8) is evidently satisfied by (0, P, Q, R], and also by 
[P,0,—S,—T]; and these solutions are independent, because the determi- 
nants of their matrix cannot all be zero unless P=O, a supposition which 
may be rejected as it implies that a=0, 7. ethat dis a square. From this 
set of independent solutions a set of fundamental solutions is deduced, as fol- 
lows. Let « be the greatest common divisor of P,Q, R; and let & be deter- 
mined by the congruences & Q —S=0,k ae T=0, mod a which are simul- 
lt # rc 
Q 
taneously possible, because — and Me have no common divisorwith the modulus, 
Bb 
while the determinant 4 (RS—QT)=—UF is divisible by it. ‘The solutions 
H rf 
[us 1, Fa— ps ER p =|, [°. = QR are then a fundamental set, and may 
P P Hep 
be taken for [p, p, p, p,], [4 9, 995] respectively. We thus find Ann!= 
aS or A= ne ; 2Bnn'=R+S8—2k =. Multiplying this equation by A 
B I B B 
QR 
—, — in succession, and attending to the congruences satisfied by k, we obtain 
# # t Ul Ul t 
the congruences P p= wh, Q Bee ab,R B= bb +Dnn’ mod A; which deter- 
# a HOB 
mine B, for the modulus A, because £ a B are relatively prime, These 
Boe pw 
determinations [viz. of A, and of B, mod A] are sufficient for our purpose; 
2 12 
because if B’=B-+4NA, the forms (4 B, 2 <) anil (4 Bi, 3 x) ate 
equivalent. To obtain, therefore, the form compounded of two given forms 
(a, b,c), (a', b',c'), we first take the greatest common divisor of d’ m? and 
d m’” for D (giving to D the sign of @ or d’); we then determine n and n’ 
