OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 487 
e“ Q,r (cos d) - l {e- (/x + 0 . i) + e 4 ”" 1 QA (/* - 0 . t)} . . (29) 
we 
find 
_ . .. /-IT (n + in) 
QA ( cos 0 ) = \/ 7r *n(?t + f)' 
I -i 73" Wl7T 
cos [ n + ^ 0 + ^ + —y 
-4m 2 C0S U + f^+-T + -o 
37r . mir \ 
(2 sin 0) 
2.2?x ~f~ 3 
(2 sin 0)® 
+ 
l 3 — 4m 3 .3 2 — 4m 2 ™s{ii + y0 + — + - 
2.4. 2n + 3. 2n + 5 (2 sin 0)* 
5tt 
mir 
(GO), 
the couvergency condition for this series is the same as for (59). 
It may be remarked that the series (57) is convergent if /x is a real positive quantity 
o 
greater than unity, (= cosh xfi) provided \Jj > b log 2, or cosh xp > -—= ; in that case 
we have 
QA (cosh \jj) 
g)MU 
y- n (n + m) e ~^ + ^ f l 2 - 4m 3 c~* 
v/7r II («- + |) (2 sink { 2.2?t + 3 2 sinh 
, l 2 - 4m 2 . 3 2 - 4m 2 _ e-W 
2.4.2 n 4 3.2 n + 5 * (2 sinh o|r) 4 
where cosh x[j > — 
2 w 2 
The corresponding series for PA (cosh i//) is not convergent. 
30. The series (59), (60) are convergent, provided 6 lies between —and 
(131), 
—; it will 
0 
now however be shown that in case m and n are real, and n 4- m — 1, \ + m are 
positive, a finite number of terms of the series will represent approximately the values 
of P n m (cos 6), Qa (cos 6) when the restriction as to the value of 6 is removed. To 
prove this, it will be necessary to estimate the remainder after any number of terms 
in the series (57). 
It has been shown by I)arboux # that if x is a complex quantity, Maclaurin’s 
theorem takes the form 
f(x) =/(0) + xf (0) -f- 
9 ! 
f" (o) + ... + 
\.f (8’x) 
where 6' is a proper fraction, and X denotes some quantity whose modulus is not greater 
than unity. Applying this result to the expansion in Art. 28, by which (57) was 
obtained, we see that the remainder after r terms of the series for Qa (cos 6 -j- 0 . i) is 
l . e 
n (m 2) n ( _ 2) e -(«+i)i 0 -ij 
1 
47t sin (m + n) 7 r 
- (- IV 
\r \ / 
IT (in + J 4 r) 
IT (r) II (m — 4) 
A2 sin 0 ’ (z 1 — 1)'' 
(0+, l + i 0—, 1—) / Q’ u \-m-h 
(1 — u) >l+,/t X ^ 1 + XAlj <lu > 
w 
* See Liouvillk’s ‘ Journal,’ Series III., vol. 4. 
