280 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
(5) If O is uneven, and A unevenly even, /'as well as F being properly primitive, 
there are 16 solutions of the congruence OF=l, mod 4; for this congruence may be 
written in the form (Art. 6) 
2 ax 2 + 2 j&y 2 -{- yz 2 ^ 1 , mod 4. 
For clearness, we shall henceforward represent by r any uneven prime dividing both 
O and A, by h any uneven prime dividing A, but not O ; by u any uneven prime 
dividing O, but not A. Let 6=2 — "F, if Q=A=1, mod 2; 6=2, if, /and F being 
properly primitive, O is uneven and A unevenly even; Q=4 in every other case; also 
let 
V=4IIrxIL y xIB, 
'W v )=|V i n[i-i]n.[i—i]n[i-i]n[i- (-//]. 
Combining the lemmas (1) . . . (5) we obtain the theorem — 
“ The form F represents numbers of the series M for \p (V) of the V 3 systems of values, 
mod V, that can be attributed to x , y , z.” 
Let x t , y { , Zi represent one of these ^(V) systems of values ; it is evident that F repre- 
sents a number of the series M for every system of values of x, y, z included in the 
formulae 
x~=- 1 
y=VY+yA . . (27) 
s=V Z+ 2 J 
in which X, Y, Z represent any integral numbers whatever. It is also evident that 
there are as many systems of values of x, y, z included in the formulae (27), for which F 
acquires a value not surpassing L, as there are points having their rectangular coordi- 
nates of the form 
T — VX + a?,- 
VL ’ 
y= Z/+L', 
VL 
g = VZ+*i 
VL 
and lying inside, or on the surface of, the ellipsoid, 
Y{x,y,z)=l. .......... (28) 
Let Vi be the number of these points, and let L be increased without limit ; the limit 
of the fraction is the volume of the ellipsoid (28), or f . Extending this result 
to all the %f/(V) values of i, we find 
Xvi 
lim —4 — 
L* 
4 44v) _ ^ 
3 v 3 ’a 
( 29 ) 
