374 MESSES. C. AND E. CHAMBERS ON THE MATHEMATICAL EXPRESSION 
according as neither (s 4-^)2 nor {s—t)z is 0 or a multiple of 2-r; as (s+£)z is not, but 
(s — t)z is 0 or a multiple of 27 t ; as ( s-\-t)z is, but (s — t)z is not 0 or a multiple of 27 t ; 
or as both ( s-\-t)z and (s—t)z are multiples of 27 t. 
Now by Bessel’s process, and assuming, as we shall, that only a few of the first terms 
of the expression for a m have considerable coefficients, 
Also let 
2*=»“i 2 m=n ~ l 
~X u m cos tmz =p t , and - X a m sin tmz — g_ t . 
™ 7 W = 0 ^ m = 0 
Q m—n—\ Q m=n—\ 
-X a m , cos tmz — P A and - X u m , sin tmz = Q ; . 
n m = 0 n m =0 
Therefore, writing a t and a* respectively for the coefficients of i and i 2 in (22), and 
transposing, 
p t =V t —a t i—SL t i 2 
Proceeding in a similar manner, we find : — 
(28) 
a m , sin tmz=u m sin tmz 
r . s 2 i 2 
+ 1 si(q s m cos smz sin tmz —p s m sin smz sin tmz ) — -x- (p s m 2 cos smz sin tmz 
> (24) 
I 
4- (pn 2 sin smz sin tmz) J ; 
a m , sin tmz =X u m sin tmz 
m=n— in 0 « 
+s If 
1=0 L _ 
S 2 * 2 
4 
q s [m sin (s 4- t)mz — m sin (s — t)mzj —J9 s {m cos(s — t)mz — m cos(s 4- t)mz) j >■ (25) 
| p s (m 2 sin(s 4 -t)mz— m 2 sin(s — t)mz) 4 - q s (m 2 cos (s — t)mz — m 2 cos(s-{-t)mz) | | 
X u m ' sin tmz = 2 a m sin tmz 
t Vsi( ( n s + t n s—t \ ( n n\) s 2 i 2 f / rt 2 s + £ 
4-| l 2 p - ^ ' “ 2 COt ~T- Z+ 2 COt ~2~ Z ) -?• V ~ 2 + 2 j { _ T \P\- 2 COt ~Y Z 
. rc 2 , s t \ * / w 2 , n 1 , n 2 n 1 \)~\ 
4- 2 COt 2 Z)+q s l 2 +2 . o ® — t 2 2 . 2 s + * )f ’ 
/ \ sin 2 — —z sm 2 ^— z / t -i 
m=n - 1 
or =X sin tmz 
!~si( ( n s + t n \ (r? n w\) s 2 i 2 ( / n 2 s + * A 
+ 2 COt_ 2 _2: + 0 j - ^V' 2 _ 2 '+ 27 } 4 {p(- 2 " Cot " 2 "^ + 0 
fn 3 »*.«,»* n 1 \)~] 
+ 6+2" - 2 . 2 s + t j f I’ 
\ sm^— z' J -J 
y (26) 
