84 
MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 
C + ^ e tan »- .1=51 
» + 1 D 
n — r — F 
cos* 
Y-X 
cos* 
X 
[n - 1] 
+ » j *+ i ) c ,, tan .- 2 
Y-X 
+ ... 
. . . + c<- » tan + !fe±I) c (.) 
Whence we obtain c' = 0, c'" = 0, c v = 0, &c. 
Also 
ST 
1 
JS, 
i— H 
1 
s 
<— 1 i 
+ 
S 
| Ji 2 ” 3 j 
(n(n+ 1) (m— 3)(m— 4f 
i Jv _ (»-3 )(m -2) 
V 4 4 J 
' ~ [m-2] ’ ' 
V 4 4 / 
! C — 4 C , 
+ 1 ) _ (” ~ 5) (» - 6) ^ cVi _ (» - 5) (» - 4) civ _ &c &c 
Consequently 
P = 
c ^ 
2* - 1 B Y - X , 
tan — o H 
ore — l 
[»] 
+ 
+ 
2 [w — 2] (2 m — 1) 
tan 
re — 2 
Y-X 
)W — 1 
2.4.[m — 4] (2m — 1)(2m — 3) 
Y — X 
2” ~ 1 tan” “ 6 — 
tan 
re — 4 
2 
Y-X 
2.4.6.[n — 6]. (2« — 1) (2 m-3).(2m — 5) 
= yT + -- 
Now 
Y-X 
= -\j- i lo sr^= tan 1 (-!*•/- 0>--- 
tan 
Y-X 
— ^ \/ — 0 ; 
whence, finally, 
p — 2 ” ~ i c ( ( ~^ ^ Z p :)n + (“M*' 3 *) 
O] 
+ 
( if - 4 
2. [m — 2] . (2 m — 1) 2.4. \n — 4] (2m — 1) (2 m — 3) 
+ 
a remarkable result, showing that in this instance P n is independent of <p. 
P being free from <p, is a perfectly symmetrical function of \jj and (i! ; and p' is a 
constant with respect to p ; therefore 
= K,( 
(— fJ. \/ — If ( — — 
re — 2 
w 
+ 
(- /X \/ -If - 4 
2 [m — 2] (2 m — 1) 1 2.4. [m — 4] (2m — 1) (2m — 3) 
=3)+-) 
(-nW-\Y 
V ■[»]' + 
2 . [n — 2] (2 n — 1 ) 
To determine K for any particular value of n, we refer to the expression from 
which the two series were deduced ; namely, 
| r 2 + r ' 2 — 2 r r' fjJ + </ ( I — (a 2 ) V ( 1 — ^/ 2 ) cos (<p — <p')) 
When (Jb and are each 1, then P re = 1, which gives an equation to find K b . 
