OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT.. 
473 
P/ J (— cos 9 — 0 . i) = e nm P,/* (cos 0 + 0. i) — " S “ 1 e m *‘ Q/* (cos 6 + 0 . i), 
77 
hence 
e *”" 1 P/ (— cos 9) 
_ _ Q-imm p m ( CO g SHI (/t + Hi) IT g _ m7rt ^ gl«Wi £ Qy« ( CO g $) - l t7r P/* (cOS 6*)} , 
or 
P,® (— cos 0) = P/' (cos 0) [e (M+m )’ r ‘ + i sin (n -j- m) 77} — 2 sin (n + m) tt ^ cos ^ 
hence we have 
2 
P« w (— cos 0) = P/ ! (cos 0) cos (n + m )tt — sin (?i -f- m) 7 r. Q/‘ (cos 0) . (33). 
77 
It is easily verified by means of the formula in Arts. 17 and 18, that when 0 = i”, 
_ 2 
(1 — cos n + ra 77 ) P/' ( 0 ) = - sin (n -j- m) tt . Q/ J ( 0 ), 
77 
hence (33) does not involve a discontinuity in the value of P™ (cos 0), as 0 changes 
from 0 to 77 . 
We have, also when 6 is between 0 and ^ 77 , 
Q, “ (— cos 6 — 0 . t) = — e nm Q/ ! (cos 6 + 0 . 1 ), 
or 
WITT r -1 3'rtlir r 1 
e 2 !Q«™ (— cos 0) + lf P n n (— cos 6) | = — e' ,m . e 2 |Q n * (cos 6) — — P B ra (cos0) |, 
hence, by means of (33), we obtain the relation 
Q/ 1 (— cos 0) = — Q.„ m (cos 6) cos (w + m) 77 — ^77 sin (?i + w) 77 . P n m (cos 0) (34). 
When m and n are real integers we have 
P v n ( — cos 9) = ( — lj n+m P n m (cos 9), Q n m ( — cos 9) = (— 1 )' rt+TO+1 Q.y (cos 0). 
Expansion of P n m (f), P n m (/x) in powers of /x — vV 3 — 1 • 
21 . If we make (/x — vV 2 — l) 2 , for which we shall write the independent 
variable in the differential equation ( 2 ), we find that the equation takes the form 
MDCCCXCVI.—A. 3 P 
