364 MESSRS. C. AND E. CHAMBERS ON THE MATHEMATICAL EXPRESSION 
which, since rz=2g‘T, and if be not a multiple of 
or ( sf-±tg ) not a multiple of r, 
=0. 
Similarly, under the same conditions, 
m=r-l * -P 
% sins-m2sintfm;z=:0; .... 
m= 0 9 
X sin s-mz cos tmz=0 , 1 .... 
> invariably. 
m=r-l S J 
X cos s-mz sin tmz=Q, .... 
m=0 9 J 
If, now, we define as follows. 
j - + {the sum of all the values ofjt?, for which s is 0, or such that sg is a multiple of r}~| 
L + { 
„ 
„ 
p e 
„ s is 0, or 
„ 
}J 
i* 
„ 
„ 
Ps 
„ s is 1, or 
„ (s+\)g 
„ 
>1 
^ + { 
» 
» 
p s 
„ s is 
» (tf+ff) 
5 
'+{ 
„ 
„ 
9* 
„ s is 1, or 
» (s-l)ff 
„ 
}"] 
_ 
-{ 
„ 
„ 
9s 
„ s is 
„ 0+1 )g 
„ 
} 
+ { 
„ 
„ 
Q. 
» s n 
» (*f-9) 
„ 
} 
’ 
1 
» 
» 
Q. 
„ s „ 
» ( s f+s) 
-1 
"+{ 
„ 
„ 
Ps * 
„ s is 2, or 
» 0 +*)9 
„ 
!, 
- +{ 
p e 
„ s is 
» 0/+ 2 .9) 
» 
r 
"+{ 
„ 
„ 
9s 
„ s is 2, or 
» 0 -9)9 
„ 
-{ 
„ 
„ 
9s 
„ s is 
» (s + 2)g 
„ 
} 
+ { 
» 
„ 
Q. 
„ s „ 
„ (sf-2g) 
„ 
} 
5 
L-{ 
» 
» 
Qs 
5» s „ 
„ 0/+%) 
} 
rc. 
&c. 
&c. 
&e. 
1 
r +{ 
„ 
„ 
Ps 
„ s is t, or 
» (s+t)g 
„ 
n 
L+{ 
»» 
» 
p , 
„ s is 
» (sf+tff) 
» 
5 
r+{ 
„ 
9s 
„ s is t, or 
» 0-05- 
„ 
in 
-{ 
„ 
9s 
„ s is 
» (s+t)g 
„ 
> 
+{ 
„ 
Qs 
„ s „ 
» (tf-tff) 
,, 
> 
5 
u-{ 
» 
Qs 
,, s „ 
» (tf+tff) 
> 
> 
r +{ 
„ 
„ 
p . 
„ s is 1 , or 
.. 0+1)/ 
„ 
n 
L+{ 
» 
» 
Ps 
» s is 
» O^+Z) 
, 
r + { 
» 
„ 
Q. 
„ s is 1, or 
» 0-1)/ 
„ 
>n 
-{ 
« 
„ 
Qs 
„ s is 
» 0+1)/ 
„ 
} 
+{ 
„ 
9s 
» s » 
» 0 9-f) 
„ 
> 
9 
L_-{ 
» 
» 
9s 
» s j> 
» 0 9+f) 
„ 
} 
r+{ 
p s 
„ s is 2, or 
» 0+2)/ 
,, 
In 
1 
» 
» 
Ps 
„ s is 
» 0 9 + *f) 
» 
1* 
