22 
Mr. Ivory on the Method of computing 
the general term being 
(l-V)t. ( . r <0 . COS. /> + (! — (*')*• (i-y)i.A (l> 
. sin. itp, 
where and represent rational and integral functions 
of y. Now if we multiply by dtp and then integrate from <p 
— o to (p = 27 r } we shall obtain as before J ’ v'dep = x ; 
because the integrals of all the terms multiplied by the cosines 
and sines of the multiple arcs are of the same magnitude at 
both the limits. Therefore, by following exactly the same 
procedure as before, we shall arrive at this equation, viz. 
ff 
(r— p) • p 2, . v . du! . dv C 
jr *— zrp . y+p 2 j L_LL 
= .V 
i— i 
(o) 
in which the function r (o) is to be valued on the supposition 
that y — 1 . But the suppositions y= — l, <p = o, correspond 
to ;j/ = — {j u, w'= w; and the suppositions 7 = i, <p = 2? r, cor- 
respond to ix' = [x, and ui'= & -j- 27 r : therefore if u denote 
what v' becomes when ;x' = >x and m' = ra- + 27 r; that is if u 
be the same function of |x, V i — jx 2 . cos. ot, y / 1 — [x 2 . sin. ts 
that f is of [x 4 * * 7 , V l — [x /2 . cos. w 7 , v/ 1 — (x 72 . sin. x? 7 ; it is 
plain, from the transformed value of v' y that u = when 
7=1. Therefore, we shall have 
ff 
(r-t) 
i— i 
| r*—2rp . y+ e 2 | l . 
p z . v . dij.' . d'Ts' 2 7r 
“ /—I 
-H 
2 
U. 
(E). 
4. The investigation j ust gone through shows how neces- 
sary it is to retain all the terms we have done in the equation 
(C), and at the same time it proves that the terms thrown 
out in finding that equation were justly rejected. It completely 
