496 
THE ASTRONOMER ROYAL’S EXPERIMENTS 
result thus obtained with respect to a, from the limit b to the limit b + c, we finally 
obtain 
X b + c-b I * 3 . 
6 / 
{— (cos/3— cos a) + (cos 3 3 — cos 3 a)} 
, b + c —b b f 
T" or>^,4 { 
80/ 
b + c —b 7 
' 896 / 
+ 5 
Y b + c 3 — b 3 
9(cos/3— cos a) 4- 33(cos 3 /3 — cos 3 a) 
— 39(cos 5 3 — cos 5 a)-|-15(cos 7 (3— cos 7 a)} 
{— 75(cos3— cos a) + 575(cos 3 (3— cos 3 «) 
— 1590(cos s 3 — cos 5 a)+2070(cos 7 3 — cos 7 a) 
— 1295(cos 9 3— cos 9 a) -f- 315(cos n 3— cos 11 a)} 
b + c 
2 {+ (sin 3 3— sin 3 a)} 
-b b 
80j/' 
+ 
+ . 
A 7 
896 / 
{ — 12(sin 5 3— sin 5 a) -|-1 5 (sin 7 3~ sin 7 a)} 
{ -f-120(sin 7 3— sin 7 a) — 42 0( sin 9 3— sin 9 a) + 315 (sin 11 3— sin 11 a)} 
These expressions for X and Y will he converging for all points situated at a greater 
distance than b+c from any point of the axis AB, inasmuch as they are composed by 
adding together corresponding terms of series which are then all convergent. Among 
other points, these expressions hold for such as are situated on the axis external to the 
bobbin, and not nearer A or B than by the distance (b + c). For such points, however, 
the expressions become illusory, assuming the form § ; they may, however, be evaluated 
by the methods for the evaluation of vanishing fractions. Y is clearly zero. X may 
be more readily obtained directly from the expression for U ; from that expression we 
find that for a single circular current the attraction on such points is 
X=2**4+3~!3 + 
1 5 
Hence, in the case of a bobbin, if x be the distance of the attracted point from O, the 
middle point of the axis of the bobbin, we have 
X 
drdal+Zs- 
«.'*+/ Ji 
I 9 b + C b . n\ ,a , 
° 40 ( a ? 2 — , /‘) 4 ) 
_ b + c —IP 
5 112 fo+/ -X~f ) 
+ 
which gives X for points situated on the axis for which x is not less than (b + c+f). 
