extensive Class of Spheroids . 6 g 
2 /3 (k) . ( i — {J-* ) ^ • (1 — p/*)* . ^ . j~ . | cos. ns . cos. w®' 
+ sin. tzot . sin. tzot' j ; 
and by combining this with the general term of y', there will 
result the following expression which is clear of the sines and 
cosines of variable angles, viz. 
(l — .fcos.nvrxjfj/'fi^ . ( 1 — {J-' 1 ) 71 . M^. dyj. dm 
d[j. ^ d/j. 
-f- sin. n~ x . ( l — p.* )* . d -~ . . df . dv ? J : 
this expression again comes under the method of No. 4 ; let 
the integral y ' . df . d-®' . Q (0 be denoted, as before, by 
27 r . ; then the part of derived from the general term 
of y, will, by the method alluded to, be thus expressed, 
/ (I — (J. ) . : . d[x. 
r ^ d 
. ] cos .ms x - — 7 — - — ? — 1- sin. nvj 
(. 2.4.0 21 * 
i av » d n C(d 
( 1—^ 
/ ( I -**'»)*. — r .dd 
dy'- n 1 # 
3 * 
2.4.6 
and if all the parts of be computed successively by means 
of this formula, the complete value of that quantity will be 
found by collecting them all into one sum. 
Having thus determined the value of the integral JJ' y' . 
df . dm ' . O^, denoted by 2 tt . U ( ^, we have 
B (i) = 
x . 2?r . a 
*+3 |j(0 . 
but it is to be observed, with regard to the case of i = 0, that 
