540 
PKOFESSOK G. H. DAIFWIN OX ELLIPSOIDAL HAPMOXIC AXALYSIs. 
1 now wish to sIioav that, in the use to which these integrals are to he put, the 
logarthmic terms disappear. 
The following is a table of tliese integrals collected from (87), (89), (90), in as far 
only as they involve logarithms :— 
IY = (1+P+iV/3') log ^. I = 1 -:]:^) log 
! A 
feus- 0 
/3 
13 
^ ^ J0=:-l3{l + i^) log J , I COS' eid6=~ I-/3• log — , 
/9 
Then the formulm (83) for L, M, Xi, in so far only as is at })resent material, are 
L 
F' 
dO. 
.51 
^ (cos- t) + /3/'cOS' ^^)-l- ^/cos^ $ 
( 7 =|[ 7 oX (cos- A/Sp'cos’ 0] d6. 
Y (cos- (cos'^' 6-Y^h cos- 6^)+^/3A cos^ 6 
X 
If 
dO, 
On using the integrals and only retaining squares of/3, we find 
E>/ 3 To (|-/‘)log 
f=^T„ 0-i)logA. 
[ — (1 + 4 ^) + ^' (1+2‘^)J+/3A(t—/<)}' og ^ • 
But by definition o^ f and g in (80) and of the «’s in (82), to the order zero of 
small quantities, 
J—d-, 9—^- /< = 1^0=0 1) = ~T/' 
Thus the logarithmic terms entirely disappear, and henceforth may be dr(.'[)]3ed. 
Thus, u.sfar as ruatcrial, we have the following tal)le of integrals :— 
