; 
_ safely infer that ts 
35 
sin ma = Ax + Ba? + ca’ + &e. 
justifying this assumption on the ground that 2 and sin ma 
vanish together ; which can be considered valid only so long 
as m = oo is excluded. In fact, whether we seek the deve- 
lopment of sin ma after the manner of Lagrange, or by the 
theorem of Maclaurin, it is essential to the very nature of the 
investigation that the unknown coefficients a, B, c, &c. be all 
assumed to be finite. We cannot conclude, therefore, from 
dsinmxz 
dz 
infinite: and similar considerations forbid the conclusion that 
dcosmz 
dx . 
limits equally militates against such a conclusion; thus, if the 
function were sin z, we should have 
sin (w +h) — sing = 2sin$h cos(a + 4h), 
Lagrange’s reasoning, that = m cosmaz, when m is 
= — msin mz, in like circumstances. The method of 
or 
sin(a# +h) —sinz  sinth 
win on ed Jarabe Th cos (x +4 h); 
and since 
1 
= 1, in the limit, or when A = 0, we should 
dg = 082: But, by proceeding in like 
manner with sinmz, we should have 
Se Sn on co (ma + mi) 
from which, if m be infinite, it could not be inferred that 
dsinmz 
= mcosmax; since we have no right to affirm that 
tends to 1, as A diminishes, and finally terminates in 
that value when h = 0; nor that, in like circumstances, 
cos (mx + ie =cosmz. We have nothing to justify the 
ni ind 
th pie sin mh 
; i Enh are the same at the limits 
assertion that * 
