32] 
TO THE THEORY OF SPHERICAL COORDINATES. 
221 
Suppose, as before, P is given by one of the systems co, p; and let «x, p 1 be 
the new systems which determine the position of P with reference to X 1} Y 1} Z x . 
In the first place, v lf are given by the formulse 
W(e',e", q) _ F (s', s', q) 
1 ^{W(e', e', q) W(e", e", q)} s', q) F(e", s", q)} * 
= W(e", e, q) W(e", s, q) 
(e". e", q) TT (e, e, q)} *J{W (e", e", q) F(s, s, q)} ’ 
F(e, e', q) W (s, s', q) 
^ V{W (e, e, q) W(e', e', q)} J{W(s, e, q) F(e', e', q)}’ 
The system is evidently given immediately by 
t . „ . y _ ^0* p q) . ^O', p> q) . ^(«", p, q) 
S ‘ ' ^ V{TV(e, *, q)J ' V{W(e', o'. q)J ' V{W(«", q))'" 
F (e, co, q) F (e', co, q) F (e", o, q) 
V{F(e, e, q)} • J{W(s', s', q)} : V{F(e", q)}’” 
and from these we may obtain the system p 1 , by means of the formulse 
Xi : y 1 : * = + 5^! + &£i : &x£x + fix^x + : qxfx + + Cxfx 
.(86) 
(87), 
(88). 
This requires some further development however. We must in the first place form 
the system 3j, bx, C x , b, be this is done immediately from the formulse of Sect. 2, 
and we have 
eV', q) A;F(eY', eV^, q) 
fll = F(e', e', q) W(e", e", q) = W(e', s', q)W(s", s", q) 
F( e"e, e"e, q) kW ( e"e, e'T, q) 
hl= F(7', «"Tq) F(e", e. q) = F(7', e'CI) F( e , e , q)' 
F ( tfe', ee', q) A; F ( ee', ee', q) 
Cl= F(e, e, q) F (e', e', q) = F(e, e, q)~ F~(Y, e', q) ’ 
f F ( e"e, ee', q) A; F ( e"e, ee', q) 
f ‘ = F(e, e, q) 7\WJT, TTqj F(e", q)} = F(<-, e, q) V(Tf (¡CPi) F (e", e", q)(' 
W («7, 7?, q) ifeF(T?, 77', q) 
9 ‘ = WTTTq) V !F(e", e", q) FfeV, q)J = F(e\ e', q)V(F(e", <r", q) F(e, e, q)j ’ 
F ( 77', 77, q) k If ( 77', 77, q) 
6l = F(e", e", q) 7(F(e, e, q) F(e', 7, q)} = F(e", 7', q)V(F( e , <=, q)F(7, 7 q)('