44] 
CONNECTED WITH THE THEORY OF ATTRACTIONS. 
289 
Note.—One of the intermediate formulae of Mr Boole [in the Memoir referred to] 
may be written as follows: 
S = - f da f dvv q cos [(a — a) v + \ qir] 6a, 
7T J o Jo 
or what comes to the same thing, putting i= -1, and rejecting the impossible part 
of the integral, 
l. e. 
8 = - el* 
7T 
S = I I(pa da, 
J 0 
Now (a—a) being positive, 
r i r°° 
a da dv v q e- v (a_tr) (pi, 
Jo Jo 
j = _ e h q in dvv q e iv (<*-*). 
7r ! n 
I = — e^ qvri r (q + 1) e^ {q+1)ni (a — a)~ q ~ l ; 
i.e 
/ = - e (?+ *)« r^ + l) (a-a)- q -\ 
or, retaining the real part only, 
/ = — — sin qir T (q + 1) (a — cr) - ? -1 
i.e. 
1 T(-q) (a ^ q K 
But (a — a) being negative, 
I = — e^ qni r (q +1) e~$ iq+1)7ri (a — a)~ q ~ 1 ; 
l. e. 
/= ^ e~i ni F(q + l)(a- a)~ q ~\ 
or, retaining the real part only, 1 = 0. 
Hence 
8 
or putting 
= rF7) 
a = a +1 (1 — a), or a — a = t (1 — a), 
8 = f 0 l ~ q ~ l + <*)] dt ; 
the expression in the text. Mr Boole’s final value is 
8= - 
d\ q 
da 
which, though simpler, appears to me to be in some respects less convenient, 
c. 
37