OF UNRESTRICTED DECREE, ORDER, AND ARGUMENT, 
489 
when (3 denotes tan 1 —; finally this remainder is numerically less than 
+ 1 / — II (n + on) l 2 — 4 m 2 .3 2 — 4m 2 .... (2 r + l) 2 — 4?n 2 1 
X n ' n (n + f) 2.4 ... 2 r. (2 n + 3)... (2 n + 2r + 1) (2 sin 6) r + * * 
it has thus been shown that for all real values of m and n such that n m — 1, 
m + ^ are positive, the series (60) may be used to obtain an approximate value of 
Q,/" (cos 9) for all values of 9 between 0 and tt ; if the first r terms of the series are 
taken, the error is certainly less than 2 m + 1 times what we get by writing unity for 
the cosine in the r + 1 th term, r being any number greater than m + A particular 
case of this theorem, namely, that in which m — 0, and n is an integer, has already 
been obtained otherwise by Stieltjes. # 
It has been shown that nre mm P„“ (/x) = e imm Q„ m (/x — 0 . i) — e“*” m Q„ m (/x + 0 . t), 
it therefore follows that the series (59) for P,“ (cos 6 ), may, under the same conditions 
as regards n, m, be used to obtain approximate values of P/' (cos 6 ), the error being 
limited in the same manner as in the case of (60). 
Approximate Values of P„ M (/x), Q n m (g) when n is a large real quantity 
and [x is real. 
31. It is well known that when n is a large integer, 
n (n) 
n (n + i) 
is approximately 
n (n) 
equal to ^=, it follows from (59), (60) that the asymptotic values of ^ (cos 0), 
f — Q/ J (cos 6) for a large real integral value of n are given by 
U(n) 
P » m (cos 0) = /\J 
nir sin 6 
IT (n + on) 
H ( ' l} Q * (cos 0) = e mm /\J — J 
sin (n + A 9 + , + f 
7T . 0110T 
2 
II (n + on) 
„t+t - ■ m 
These expressions are generalizations of the known asymptotic values 
P„ (cos 0)=f sin (» TTJ + j) 
which was given by Laplace, and 
Q„ (cos 0) = \f cos (“+T® + "j) 
given by Heine.! 
* 1 Annales de la Faculte des Sciences de Toulouse,’ vol. 4, in a paper entitled “ Snr les Polynomes 
de Legendre.” 
t ‘ Kngelfunctionen,’ vol. 1, p. 175. 
MDCCCXCVI.—A. 3 R 
