80 
MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS* 
and by repeating a similar process* we obtain 
^ +#3ta„^ + ? *- 6 
dXd Y ^ dX 
4 cos 2 * * 
X-Y 
= 0. 
By observing the assumptions here made, and the results obtained* we find that in 
the first assumption = v), the coefficient of P re tan — ^ — is 0; in the second, 
X Y 
that of v tan — ^ — = 1 ; and so on, in the order of the natural numbers ; and in the 
results, the numerical coefficients are 1 , 2 , - 4 ~ " ; 3 , ... generally/*, - — ^ — “• 
I shall prove this in the general case, by showing that if it is true of one value of n 
(as we see it is), it is true of the next value, and so on. Let the ( n — l)th substitu- 
tion give 
. I? (n _ n tan X ~ — 4 - _ 0 
dXdY + dX^ n ^ tan 2 + ? . n X-Y “ 
and let 
then, as before, 
d 2 
dXd Y 
4 cos~ 
dg . , „ X-Y 
+ (n — 1 ) g tan —j— = s : 
g , t ^ \ 4 . X — Y dg n — 1 
— + (n - 1) tan • jj, + —5- 
cos z 
X-Y - 
ds , 
dX 5 
and, therefore, 
d s 
dX 
, a — (n — l)n ^ . d s 4 
+ § X - Y — 0 , and § — — ^ 
X-Y 
4 cos 2 
, , x cos- — r- 
(«—!)« 2 
a g 
d 2 s 
d Y — dXrfYa-(»-l)» 
Consequently 
cos 2 
X-Y . X-Y 
4 cos — - — sin — - — 
, X-Y _ ds_ 2 2 _ 
2 d X a — (« — I) « 
d 2 s 
COS 2 
dXd Y a — (ii — 1 )n 
ds 4 (n — 1 ) 
X-Y ds 4 
2 dXa — (n — \)n 
X-Y . X-Y 
COS — — sin — r — 
dX a — (n — 1 )n 
X-Y . X-Y 
cos — s — sin — r, — = s, 
or 
d 2 s 
dXd Y a — (n — l)n 
that is, 
cos 2 
X-Y 
-f 
ds 
4 n 
dX a — {n—\)n 
X-Y . X-Y , 
cos — s — sin — X f- s = 0 ; 
dh 
ds 
dXdY dX n tan 2 1“° 4 2 
and, therefore, the law of coefficients, as above stated, is correct. 
X-Y , a-n(n- 1) „X-Y 
4 - s - — 
cos 2 
0 , 
