82 
MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS 
Such is not the case. At each integration a function of X must be added, and these 
functions determined by reference to the original differential equation*. 
7 . Returning to the value of P w , we have 
p„ = xi(f- k'"- 1 Io & f=f) + 4 (*> + 7 ' / ~ l lo s r=f)- 
Now in the calculation of attractions, where P 0 is the coefficient of r° in the ex- 
pansion of 
j r 2 t J 2 _ rr > -j- v" (1 — yr) */ (1 — yJ 2 ) COS (i p — <p')^ | 
we know that it is 1 ; consequently 
& (f - W-^ llo sr=i) +'K?+ ^ Zll# 5rrl) = 1 ’ 
and expanding by Taylor’s theorem, we get 
*, * - * §V= T log ^ ([log L±^) 2 
+#^ (w^:) 3 +&e. 
+ * , + * , I ^ - 1 log « (A log 
-s^a*og^-:) 3 +& c. 
By equating the coefficients of the same powers of log ! , we have -(- 4 <P 
= 1 , and 4 / <p — x!i P = 0 , or 4 <p — <p = constant. 
Therefore 4 <P and %,i <P are absolute constants, and their sum is 1 ; whence it fol- 
lows that % <p = 0. Let 4 4 ~ o? then 
> = 1. 
n 7 /- _ 2 Y -Xf _ 2 Y — X /"* 1 
p„ = c v — j cos — 2~v cos —2~J ~2 
cos 
_ 2 Y - X 
d Y d Y . . . (n times)^. 
Effecting these integrations, we find that P w consists of a series of powers of 
* The common differential equation (1 — p-) 4T? — 2 p + n (n + 1) P„ = 0 will illustrate this point. 
d p°- dp 
Let P„ = 
d- z 
d z 
, and after substitution, differentiate n times ; then (1 — u~) — 2 {n + 1) p - — = 0, whence 
dfi—n dp- dp 
z = C ^ d p q- m . It is clear that no more arbitrary constants than k and m can be introduced ; and 
J{\-p a -) n + l 
yet if the integrals were left indefinite, we might obtain an integral of an expression which should differ from 
the integral of the same expression obtained by a slightly different process, by a constant. By another inte- 
gration this would cease to be a constant, and we should obtain thus different values for P w . The fact is, that 
constants must be added at each integration, and recourse had to the original equation, to determine them in 
terms of m, k, and p. 
