216 
ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION 
[32 
Imagine now a line AO, and let a, (3, y be the cosines of its inclinations to the 
three axes. Suppose also, 6, </>, % being its inclinations to the coordinate planes, 
we write 
sin в 7 sin ф sin у 
V(a) ’ V(b)’ V (c) 
(25). 
If we consider a point P on the line AO, at a distance unity from the origin, 
we see immediately, by considering the projections in the directions perpendicular to 
the coordinate planes, that the coordinates of this point are a, b, c. By projecting on 
the three axes and on the line AO, we then obtain the equations 
a = a 4- vb + /¿с (26), 
/3 = va 4- b + Ac, 
у — fxQ/ -b A& 4" c, 
1 = m 4- /36 + у c (27), 
from which we obtain 
kct = йcc 4- j)/3 + 0y (28), 
kb = i)a + fa/3 4- fy, 
kc = 0a 4- f/3 + Cy, 
l=aa+/3b + yc (29), 
and hence 
1 «= a 2 + b 2 4- c 2 4- 2\bc 4 2¡xac 4- 2vab (30), 
к — Дсс 4" Ь/3 2 4" Су 2 + 2f/3y 4- 20ay + 21)я/3 (31)- 
Hence writing 
a, b, c = t (32), 
a, & у — T (33), 
1, 1, 1, A, fi, v = (\ (34), 
a, b, c, f, g, b = q (35), 
we have the equations 
l=W(t, t, () (36), 
*=Ж(т, T, (I) (37). 
Let AO' be any other line, and 8 its inclination to AO: a', /3', y', a', 6', c', the quantities 
corresponding to a, (8, y, a, b, c, and similarly t', t to t, t. We have of course 
1 = W(t', t', q) (38), 
k= W(t', t', q) (39).