ioo Mr. Ivory on the expansion 
tioned. One thing is indisputable. Since the terms of the deve- 
lopement contain no quantities, except such as arise from com- 
bining three rectangular co-ordinates, it follows, that when 
/(0, <p) is not explicitly a function of [x, V i — jx 2 . sin <p, 
V 1 — jx\ cos <p, it must be considered as transformed into such 
a function. Algebraically speaking, the transformation is no 
doubt always possible ; but there may be danger that, by 
proceeding in this way, we fall upon expressions which are 
not proper representatives of the given function ; which are 
symbolical merely, and which cannot be safely employed in 
the investigation of truth. 
In order to fix the imagination, and to avoid every sort of 
uncertainty and obscurity, I shall take a particular, although 
a very extensive case of the general expression. I shall sup- 
pose that/(0, (p), or more shortly y, denotes a rational and 
integral and finite function of the four quantities, sin 0, cos 9, 
sin o, cos <p. We may then substitute for the powers and pro- 
ducts of sin <p and cos <p, their values in the sines and cosines 
of the multiples of the arc; by which means we shall obtain, 
y = M^+ cos<p cos 2 <p + &c. ; 
sin <p -f- sin 2 <p 
the symbols M^, M^,N^ &c. standing for rational and 
integral functions of cos 9 and sin 9, or of (x and V i— { x 2 . Again, 
every even power of V\ — [x 2 is an integral function of p; 
and every odd power is equal to a similar function multiplied 
by VT~- p : the value of y will therefore be thus expressed, 
y=[ F M++ 1 “f^ 2 •/( )) + ( F (#*)-{- V^i— p.f M ] cos cp -f-&c. 
+ (G (^) -f-V i— [x l . g ( Ip)) sin <p 
