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

210 APPENDIX.
(s) jy = sin cv.........(........ ).
Assume y = A, + A, 'v + Av.............(2),
then from (1) we have
- sin = -4, - A4v - AJ2M......(3),
and from (2) we have
d2y
1 ~2 A, -t- 2.3 A~lx: t- 3.4 A~cV2......(4).
d A2+23-3V+3;A
Now AO=y when cv = 0;..O 0;
and A, dy os when v = o;..
dcv
and therefore, comparing (3) and (4), we have
1 A3 1
Ad =0, A3= 4, A 4=o, A 5 4 ]-...
X X" X'
sinc - + &c.
ri 13 Es
(4) if we differentiate the result just obtained, we find
X2 X4
coscv= l - -+-. &c.
172 174
(5) If tan y = a tan x, to expand y in powers of cv.
(6) If sin y = a sin (y + cv), to expand y in powers of cv.
(LI.)
(1) To expand cot v in powers of v.
cot (v = c= when cv = 0; therefore there are negative powers
in the developement, and Taylor's series fails. We may
proceed as follows: