366 Concessions to Mr. Ivory. 
its linear amount is no more than the thirty thousandth of an 
inch. But in order to examine whether this error belongs to 
the original series, or to Mr. Ivory’s supposed improvement, I 
shall now follow the steps of my last postscript with still smaller 
portions of the curve, computing the value of s for x=:25, 
w='28, and r=*30, in succession. We shall then have, in the 
first place, when L is -4160, for r=+25, s=+38086, w="92463, 
2='41201, and y=-032756, as before. In the second, place, 
taking Av=-03 instead of -05, we easily obtain, from the for- 
mer computation, As=+15083 +°03077 4-:00500 + -00085 + 
[-00017], and s=-56848: the angle being about 34° 39’, the 
cosine u='$2279, and the tangent 4=-°69100. The new value 
of y may he obtained, with sufficient accuracy, from the original 
series, which is more convergent for the ordinate than for the 
sine, and it will be found y=+04830. With these values we pro- . 
ceed to find A=7‘630, B=118-2, and C=2461°2, which are 
sufficient to: make s='5685 +:1526-+.:0236 + 0066 + [6022] 
='7513 + [0022]: so that this computation fully confirms the 
suspicion expressed at the end of the postscript, that the depres- 
sion -004160, instead of being too small, is somewhat too greut, 
for a tube six tenths of an inch in diameter. 
Until therefore Mr. Ivory shall condescend to point out some 
error, either essential or accidental, in this computation, J must 
still be allowed to assert, first, That the series, with the assist- 
ance of the Taylorian theorem, or of Taylor’s theorem, by all 
means, if Mr. Ivory likes the name better, though somewhat in- 
convenient for calculation, is still both true and sufficient: and 
secondly, That Mr. Ivory’s approximations, professedly leaving 
out some small quantities as inconsiderable, are unsatisfactory, 
because they afford no ready means of appreciating the utmost 
possible value of the quantities so neglected: and because it ap- 
pears from these computations, that their very able author him-. 
self has in fact much underrated the importance of these quan- 
tities. 
On the other hand I must admit, that the accuracy of the 
series, in its original form, appears to me to have been some- 
what too highly appreciated by the author of the article ConE- 
sion : and that if the further prosecution of the inquiry were of 
any material importance, it would be right to employ a profes- 
sional computer to enlarge the number of the co-efficients, un- 
jess indeed Mr. Ivory’s ingenuity and experience could point out 
some less laborious method of determining them, than that which 
arises from the direct solution of the problem, by the method hi- 
therto employed. 
LXXVII. The 
