378 
Now multiplying both sides of (5) by Q, or w+ ix + jy + hz, 
we have 
Q™) = wW,- Vy (+ y+ 2) + (WVn + Wr) (ix + jy + hz). 
Comparing this with (6), we have 
Was =W W,, -7 Vn 
Van = Ww V, + Wr 
a pair of simultaneous equations of finite differences. From 
these we readily deduce 
Wass — 2w Waa + (w + 7?) W,, = 0, 
Ving — 20Van + (w? +7?) Vz, = 0. 
The complete integrals of which are 
Wr=c(wtry -1)"+e(w-ry —- 1)’, (7) 
V,=b(wt+ry -1)"+0(w-ry - 1). (8) 
c, ¢, 6, b', being arbitrary constants. 
«To determine the values of these constants let n = 0; 
whence from (5) it is evident that W,=w, V,=0; values 
which, substituted in (7) and (8), give 
l=c+c¢, 
O=b48. 
«‘ Again, let x =1, whence substituting in (5) for Q its 
value w + ix + jy + kz, we find W, = w, V, = 1, and employ- 
ing these values in (7) and (8), we have 
w=c(wiry -l)+e(w-ry-1), 
l=b(w+ry -1)+0(w-ry —1). 
From these four equations, we find 
[yeaa haan 1 A Ob deal 
Ovi) vd Poon? Salt z Ory - 1 
Substituting these values in (7) and (8), and the resulting 
values of W,, and V,, in (5), we have 
mE pn 
} where 7? = 2° + y? + 2’, 
C= 
—1)" =~ —l)y f 5 
+4 (ws ry Or ary a (ix + jy + kz). (9) 
