“ came its firmed fupport; inafmuch as the attraction o f the
“ Pyrenees, which by their vicinity produced a deviation in the
“ plumb-line, was the caufe o f the error. But to place in a ftill
“ ftronger light the reaions that militate in favour o f furfaces o f
“ the fecond order, I imagined it would not be deemed foreign
“ to our objed, if I fubjoined here iome general refledions con-
“ cerning the developement o f any fundions whatever in feries.
“ In this view, let | u be any fundion o f x, y , z , a, b, &c. the
“ queftion is to develope in a given fucceffion in reference to the
“ powers o f a, b, &c, and u be what we fuppofe u becomes when
“ it comprehends a, b, & c ad o.
Farther, let a = U X a A j Q x u’A ^ x abA g X &c,
X H 0 1 X « A ] ] X j X &c,
X. ¿ 'A o_2 X ^ A L2 X &c,
I b3AQ s X &c,
“ or the term whofe co-effident ambK is reprefented by A , we then
J mjf
,m x n
“ know that A — ------—— j.
' - '
1.2.3. .ffi.1.2.3,r..n
“ Now, to Ihew the application o f it to the prefent queftion,
* let us fuppofe that u — O is the equation which without any
“ approximation reprefents the furface o f the earth ; but fince
“ we learn from all the known meafurements o f degrees o f the
“ meridian, that whatever be the figure o f the earth, it does not
“ differ
“ differ fenfibly from that of a fphere, the equation u -r 0 will
“ neceffarily contain one or more o f thole fmall quantities a, b,
i &c fo affeded, that if we fuppofe them = 0, the remaining
“ equation u nr 0 will be that o f a iphere, whole dimenfions are
“ equal to thole o f the earth ; whence it will.follow that u will
“ be the firft term o f our developed feries, as we havejuft faid.
“ Befides, if we except , and ¿A o .l, all the other terms
“ a o A may be difregarded on account o f the fmallnefs o f
■ iri.n (
“ the quantities a, b, &c. ; confequently the furface-of our globb
I will be reprefented by the equation U x aAl0 x M o.
“ T he whole then comes to this, to know what is the form o f
I the fundions A , 1 and A n ,, which being fuppofed o f the fe-
1.0 o.>
“ cond order, the equation u — 0 will reprefent an -ellipfoid,
“ whofe eccentricities o f the equator, and o f the'meridian which
“ paffes through the great axis o f the equator, will depend on the
“ quantities a, b, &c. In refped o f the figure o f revolution, i t
“ is very clear that then the quantity a will be nearly equal to b,
“ or what is the fame thing, i f we make a — b _X' b, the quan-
“ tity u will be very fm a ll; whence i t follows that the equation
“ u — O, may be reprefented by this 0 U X bA . In ihort,
“ there is no reafon to fear that the difregarded quantities ihould
“ ever prove confiderable enough to produce any error o f confe-
“ quence, as difficulties which in all probability we fhall never be
“ able fully to furmount, will for ever prevent our precifely know