•<5 8 Mr, Ivory on the Attractions cf an 
mentioned by the actual expansion of the two radicals, and 
likewise determine the corresponding parts of by means 
of the formulas in No. 3 ; the comparison of the equivalent 
expressions will determine the values of the coefficients re- 
quired. 
To execute the operations alluded to, let c denote the number 
whose hyperbolic logarithm is unit ; then 
jr* — 2 ra . cos. p a* J~ s = (r — a . ~ 1 j -s . (r — a . 
c 
and if we represent the expansions of the two binomials by 
the serieses 
r S 1 rH-S 
+ A«.^- +& c. 
A + A (,) . i 
r ~ 1 " ' r 2 + i 
we shall obtain the expansion of the radical by multiplying the 
two serieses : let p and q denote the ranks of any two terms 
in both serieses, then the part of the expansion derived from 
the multiplication of the aforesaid parts, will be 
.(*> A (?) + 
r P + q + 2S 1 2 j ’ 
or, 2A tf) . A <?) . • cos. (p-q) . f. 
When i — n is an even number, we have only to make 
p -J- q = /, and p — q = n, and s = \ ; and we shall get 
r .2 C . .7 — n — j 1. a. e. I 
2 X — r X — 24 — . COS. tl<b. 
2.46.... 1 — n 2.4.6.... i-\-n r) 
for the part of the coefficient of -dk™, or of Q^\ which is mul- 
•J- i I 
