154* Ivory on a new Method of deducing 
cos. id = cos. m + iB: we may therefore suppose in series, 
u ~ sin. u = _ M<‘\ ^ ^ f + &c. 
u' — sin. u' = | . 4‘ + &c- 
where = m — sin. m\ and &c. are coefficients 
derived from : hence, 
e = B + M ~ + &c, : 
and by squaring and neglecting the fourth and higher powers 
of B, we get, 
B* = 
M< 
but, from the theory of series, we get 
2 ^(i) d .(m — sin. i — cos. m — cos. m 
d . cos. m sin. 7n ^ i -f- cos. m ’ 
therefore 
I 4- cos. m 
1 — cos. m 
.-cos.m — C’y substituting the value of 
cos. m) — 1 . Let sR* + sR'" =: (R -j- R')* 
(R — R')*; then we may substitute A for R + R', since quan- 
tities of the fourth and higher orders are neglected in the value 
of B* : consequently we shall have 
B* = ^ — r. 
But, when quantities of the fourth order are neglected, the 
final equation of the foregoing method will become 6* = ~ . 
a • 
and B, R, R', A are what b, r, r', a become, when p = o; thus 
it appears that 6* in less than when p ^o. But as p in- 
creases, 'at least after a certain limit, 6* will increase with- 
out limit, and ~ will decrease without limit; so that, when 
