218 ON SOME ANALYTICAL FORMULAS, AND THEIR APPLICATION [32 
A, M, N are its sides, l, m, n are the perpendiculars from the angles upon the 
opposite sides. Let P be the point where the line A 0 intersects the sphere: the 
position of the point P may be determined by means of the ratios f : y : £, supposing 
77, £ denote quantities proportional to the a, ¡3, 7 of the preceding section, i.e. 
£ : 7] : £ = cos PX : cos PF : cos PZ (54) ; 
or again, by means of the ratios x : y : z, supposing x, y, z denote quantities proportional 
to the a, b, c of the preceding section, i.e. 
sin Px sin Py sin Pz _ N 
x : y : z = . - ^ ^ ^ (55), 
J sm A sm F sm F v 
(Px, Py, Pz are the perpendiculars from P on the sides of the spherical triangle XYZ). 
These last equations may be otherwise written, 
x sin X _ sin PZY 
ysmY~ sin PZX W 
y sin F _ sin PXZ 
IsXYZ ~ sin PZF’ 
z sin Z _ sin P YX 
x sin X sin P YZ 
The ratios £ : 77 : £ or x : y : z, are termed the spherical coordinate ratios of the 
point P. The two together may be termed conjoint systems: the first may be termed 
the cosine system, and the second the sine system. The coordinates of the two 
systems are evidently connected by 
| : y : £ = x +vy + yz : vx + y + \z : px + Xy+z (57), 
or x : y : z = af + ib? + c£ : f)£ + b?7+f£ : + iy +f£ (58). 
The systems may conveniently be represented by the single letters 
® = & V> £ (59), 
P = x, y, z (60). 
Fundamental formula of spherical coordinates; distance of two points. 
Let P, P' be the points, 8 their distance, co, p the conjoint coordinate systems 
of the first point, &/, p' of the second; we have obviously 
W(p,p',f) 
f{W(p,p, q)W(p', p', q)} ^ 
, f{W(pp\ pf, q)} 
bm ~ Y{W(p, p, q) W (p' } p', q)} ’ 
C ot a = . 
f{W(pp',pp' q)}’