26 Prof. Powell’s further Observations on M. Cauchy’s 
L 
= S {F (m, 1, «) « > (A, 7 cos) }, 
&c. = &e. 
Whence we should derive 
4s = s{ (F (m,7,«) + F (m,r, B) +, &c.).$ (kyr, cos) } 5 
Hence by the same process as that employed before, we may 
obtain a corresponding abridged expression 
‘hee Tae ) x 
(—) =8}@)(— Sx 
e eG 
2 
To perceive more clearly the difference between the exact 
and approximate expressions, we may first observe, that since 
we have from equations (19.) and (45.) 
k 7 
ne and 7 cos 8 = Ag, 
E Sei s Ao. 
the arc which is involved in the formula becomes - el 
Now, if we take the simplified formula, develope the sine 
in terms of the arc, and divide by the arc, we shall have 
Thee 1 ;Ag.ny? WR Ne oe ‘ 
eee ee easy | 
whereas the exact formula, in the same way, would give 
1 wate 1 S[(H") (M7) 7 
aire. {1-2 = S(EE) 
oe) 
120 ee pRtee 
supposing the series to converge rapidly enough. 
Now, this would manifestly be the same as the last if we 
were at liberty to suppose 
spans (287)"] = psaay) [227], 
and similarly in the other terms: in which case we should 
