1819.] On the Maxima and Minima of ‘Quantities. 193 
tables in store, and supply on moderate terms those officers with 
them whose duty it may be to determine the longitude of a ship 
at sea, it would, in my opinion, render the working of a dunar 
observation considerably more correct, more general, and less 
laborious than at present; for a person with a very moderate: 
knowledge of arithmetic might learn the use of such tables and 
the above-mentioned rules in a few hours. 
=n y 
Errata in No. LXIX, Sept. 1818. 
Page 205, line 16, in the numeratos, vead 
2" (a+ 6x°P —(m— a) fac 
Page 206, line 2, read +. + a dtc pisses 
Page 207, line 26, in the numerator, read.(a + ¢2x")"** x d 
Page 208, line 2, in the last term of the numerator, read 
(a +. 2")” 
‘Page 208, line 7, in the last term of the numerator, read 
(@ + c2")” o 
Page 208, line 18, read = I. 
In pages 204 and 205, change Formula into Formule where 
necessary. 
ARTICLE V. 
On the Maxima and Minima of Quantities (by common Algebra). 
By Mr. ‘Vhomas Slee. fogp « 
Terril, near Penrith, Jan. 13, 1818. 
Tue design of the following paper is to enlarge the province 
common algebra, by extending it to the maxima and minima 
of analytical functions. Shouid it appear to you to be new, and 
of sufficient importance to mezit a place in the Annals of Philo- 
sophy, the insertion of it in a future number will much oblige 
THOMAS SLEE. 
NS 
Lemma.—Let y be any function of a variable x represented by 
f x, and admitting of a maximum or a minimum; then in the 
equation f x = y, x has always two different affirmative values, 
except when y is a max. or a min. and in that case these values 
are equal to each other; or if m denote the greatest or least 
value of y, the equation fx = m has two equal affirmative roots. 
The truth of the lemma may be shown in the following man- 
ner. Every function of a variable may be represented oy the 
ordinate of a curve. Let then M D M’ be the curve of which 
the equation is fz = y, the abscissa A P being denoted by x, 
and the ordinate P M 4 y, and in fig. 1 let D C be the greatest 
Vou, XIII. N° Til. N 
