An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

APPENDIX.

209

(G.)
(1) Let y = sin-,;
dy 1 I.34
then d- + - + 2- 4
doo V/!, ^2 2. 4
---- ' + &c.;
'2 4.4.. 2n
dl+' y 1... 2n - 1
d —. = -------. rFn + powers of r
d a'+1 2.4... 2n
1.3...2n- 1
=- In, when, = 0.
2.4... 2n
Hence the general term of the expansion of y in powers
of,v is
1. 3... 2n - l 2'L 1
2.4... 2 n + 1
and.. since sin-l,== mnrr when O= 0, we have
a _v' 1, 1. 3aI,
sin-a == nr + + -... &c.
1 o3 2.4 5
(2) We may often obtain developements of this kind
more simply in the following manner.
Assume y = A + SAx: + A~'v2... &c.,
dy
and A.' + 2....I + 3A3V <
Comparing this with the above value of we find
dr
immediately
1.3
A, = 1, A2 = o, 3= o ~,, = o, -... &C.
Also.A = the value of y when vG = o, which is mnr
hence y = m7 + - + - + &c.... as before.
1 1
14