488 
DR. E. W. HOBSON ON A TYTE OF SPHERICAL HARMONICS 
where X is a quantity whose modulus is less than unity ; suppose r so great that 
r — on + i is positive, the integral may then be replaced by 
fl / 6'u \~ m ~i 
— 4 sin (on — -g + r) n sin (oi + on) tt\ u~’"~ i+r ( I — u)' l+m \ f 1 , 
O -1 
z~ — 1 
du 
where the integral is now taken along the real axis. We have now 
9'oc id'u . e~‘ e _ 1 6'u 6'u cot 6 
1 + z 3 - 1 = 1 “ 2 sin 6 ~ 1 2 ~ L 2 ’ 
f / 6'u\ 2 , ] - 
the modulus of this expression is U 1-— ) + ^9^u 2 cot 2 9 1 , which is always greater 
than (1 — \ and therefore always greater than ^ ; it follows that the modulus of 
1 + 
6' 
u 
Z 2 — 1 
2 
—m—i 
is always less than 2 m+i , hence also the modulus of X 1 + 
6’ob \-m-i 
6'u \-m-i 
- 1 
is 
always less than 2 m+i ; put X (1 + , ^ 
functions of u, and p < 2 m f - for all values of u ; we have then 
* + r (l — u)" + m X (1+y—^r ) du 
= p (cos y + L sin y), where p, y are 
r« 
J 0 
-»« — £• 
Z 3 - 1 
[ u~ m ~ i + r (l —u)’ l+m p (cos y+t sin y) doj,, 
J o 
in this integral the real part and the coefficient of t in the imaginary part are each 
less than 2 m + * [ u~ M ~ i+r (1 — u) n+m du, hence the modulus of the expression is less 
J 0 
than 2 m + l \ u~ m ~ i+r (1 — u) >l + m du. Now the r -f- 1 th term of the series (57) is 
obtained by putting 9' = 0, X = 1 in the expression for the remainder after r terms, 
it has thus been shown that the modulus of the remainder after o • terms is less than 
2 m + l times the modulus of the r -j- 1 th term, and this is true for all values of 9, 
not merely for those for which the series is convergent. The two quantities 
e~ mm Q n l (cos 6 -f- 0 . i), e~ mm (cos 6 — 0 . l) are conjugate complex quantities, 
hence the remainders after r terms in the series for Q/' (cos #-|-0 . t), Q,/" (cos 9—0 . l ) 
are of the form 
(X±bY)d 
II ( —|) !!(«. +Mi,) cT (n + J)i#Tur/4 
U(n + i) 
(2 sin 6)$ 
(-i Y 
1-—4m s . ■. 2?-—1|-—4m s c- 
2.4 ... 2r . 2n + 3 ... 2n + 2r + 1(2 sin 6) r 
when X and Y are each less than using (29) we now see that the remainder 
in the series (56) for Q n m (cos 9), is of the form 
(X* + V)Cv^.§^±-^(-l) 
r l 3 - 4m 3 . 3 2 - 4m 2 ... (2r - l) 2 - 4m 3 
2.4... 2r . (2oi + 8) ... (2oi + 2o ■ + 1) 
i , 2r + 1 a \ 2r + 1 , onw a 
cos ( n + - 9 + - 77 -\—- p 
(2 sin 9) r+i 
