OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 525 
2 n cosh cr. P, ; _j w (cosh cr) — (n — m + |-) P ;;+ /"(cosh cr) — (n + m — |) P„_ i *'(cosh cr) = 0, 
with a similar relation for the Q functions, and 
P„_j w+3 (cosh cr) -j- 2 (m + 1) coth cr . P„_2" +1 (cosh cr) 
— (n — m — ^) (n + wi + |) P„_f" (cosh cr) = 0, 
with a similar relation for the Q functions. Formulae similar to these have been 
employed by Hicks to calculate the functions successively. 
58. It is important to have series for P ?! _f" (cosh cr), Q„_i w (cosh cr) in powers of 
e~ a , so that the values of the functions may be calculated approximately for consider¬ 
able values of cr. The required series for (cosh cr) is given at once by (35) ; we 
thus have 
Q„-f“ (cosh cr) 
= (— 1)'" 2 m ^ 4 —— sinh“ cr. e~ (n+m+i)<r F (m + n -f- m + n -j- 1, e~' z,T ), 
11 (n) 
and in particular 
(cosh cr) = 
n(, ‘ ^ k) F(i. n + i, n + 1, e'*). 
This is the expansion in powers of e <r , of the elliptic integral to which 
dw 
J 0 
y/ (cosh a + sinh a cosh w) 
is reduced by means of the substitution cosh cr + sinh cr cosh iv = cosec 2 6 . e*. 
The corresponding series for P /( _f" (cosh cr) must be obtained from (36), which 
requires, however, in this case modification. We observe that in the formula 
P/ (cosh cr) 
C,m Sil1 ( n ~t m ) 7r 
COS 117T 
+ 2 ** 
II (n -j- hi) 
n(»+bn(-f) 
n (n — \ 
— sinh^cr e ( H+m+1 ')* F (w+J, 
II (n — m) 11 ( — J) 
- sinh w cr . m '> <r F (m + 
n+m-f-1, n-\- f, e ~ a ) 
m — n, ■§ — n, e _2t7 ) 
when n — \ is a positive integer p 0 ; the second series has after a finite number of 
terms, infinite coefficients, moreover the coefficient sec rnr of the first series is infinite. 
The expression for V n m (cosh cr), gives us, therefore, first a finite series 
