290 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
mod 8. The congruence xz— mod 8, in which only even values of x and z are 
to be admitted, is satisfied in the former case by 16, in the latter by 48 systems; i. e. in 
either case by 
r n^-i f 2 — i | 
|Xr 2 -L 2 +(“ 1) 8 + 8 Jx2 9 systems. 
The formula (45) results immediately from the combination of these determinations 
relative to the primes S, &>, r, and 2. 
Art. 20. Let x { , y { , z { be one of the <p(V) systems of values of x , y, z, mod V, defined 
in the last article; and let us decompose the sum T of art. 18 into <p(V) partial sums 
T,, T,, . . ., comprising in the sum T, all those terms of T in which x, y , z are of the 
linear forms 
^=VX-(-^ 
•y=v Y+y* 
z=VZ-f-Zi, 
X, Y, Z denoting any integral numbers whatever. The sum Y t is equal to the number 
of points having their positive rectangular coordinates of the forms 
X= vx+ a 
v/flL 
y= 
VY + yi 
VZ +Zj 
vnL’ 
and lying within the hyperboloidal cuneus, bounded by the planes y= 0, x=z , x=2 y, 
and the hyperboloid xz—y*= 1 ; points lying on the hyperboloidal boundary are counted 
as lying within the cuneus; points lying on its plane boundaries are counted as \ each, 
and points lying on the intersection of y = 0 with x=z, and with x=2y respectively as 
^ and Let Y be the volume of the cuneus, and let L be increased without limit ; we 
have 
T 3 V 
lim L ' = a*x- s 
and since this limit is thus ascertained to be the same for all the partial sums Y„ Y 2 , . . , 
or, which is the same thing, 
Hmp=n«x ? 0 1 xV, 
1*2 V 
iimg=Apxn«x^xv. 
The value of V may be determined by dividing the cuneus into laminae parallel to the 
plane of xz; if A be the area of a section at a distance y from that plane, we find 
A=(l+/)[^log(l+^ 2 ) - log 2y ] — ^(1 — %y 2 ) ; 
whence 
