D I F
therefore, the danger lies on the other hand ; for, when all are not obliged to eat in meffes, fome will be apt to lay out their pay upon ftrong liquors, and to fquander away in one day what is but barely a maintenance for a week j but, on the fuppofition that every man contributes his mare to a mefs, we may be aflured there can be no errors in diet of any con- fequence, whilft almoft the whole pay is fpent upon common food. As to the abufe of fpirits and fruit, foldiers^are. gene- rally blamed without any foundation ; fpirits being rather be- neficial than hurtful to them, fo often expofed to the extremes of heat and cold, to moift and bad air, long marches, wet cloaths, and fcanty provifions ; and as to fruit, a kw difor- derly men may rob orchards, which is the only way they can come at fruit, but the camp-difeafes are incident to the moft regular equally with them. Sec Pringh, Obferv. on the Dif- eafes of the army, p. 86, feq.
A fundamental rule in regard to the diet of foldiers, is to ob- lige them to eat in meffes, by which means their pay will be laid out upon wholefome food. The greateft impediment to mefling are the wives and children, who muft often be main- tained on the foldier's pay ; in which circumftances it is not improper food, but the want of it, that may endanger a foldier's health.
The mefling being eftablifhed, there remains only to fee that the men be well fupplied with bread, and that the markets be ib regulated that the traders may have encouragement to come to the camp, and the meiTes have good provifions at a mode- rate price j and particularly vegetables, which, during the hot weather, ought to make a great part of the diet. In ertablim- iiig the- mefies, fome regulations might be made with regard to an allowance of fpirits, either by ftoppages on the pay or otherwife. This is already pra&ifed in the navy, and pro- bably for the fame realbns that make fpirits neceffary for fol- diers ; fince in fhips men are liable to diftempers from moift and corrupted air.
As to the diet of officers, their chief rule, in fickly times, is to eat moderately, avoiding all furfeits and indigeftion, and ufing wine in moderation. Id. ibid. p. 112, feq. DIFFERENTIAL equation is ufed by fome mathematicians for an equation involving infmitcfima] differences, or fluxions. Thus the equation 3 x * dx — 2 ax dx -f- ay dx — Sy L dy-^-axdy =zo in the foreign notation, or ^xxx—iaxx-\-ayx~2y r y -}-«#> =0 in the Englifh notation, is called a differential equation. But thefe equations mould confidently with the Englifh or Sir Ifaac Newton's notation, be rather called flu- xional equations.
Hence fome of our mathematicians have applied the term differential equation in another fenfe, to certain equations de- fining the nature of feries's. See the article Series, Append. Differential method, in mathematics, an appellation given to a method of defcribing a curve of the parabolic kind thro' any given number of points.
This method is given by Sir Ifaac Newton in the fifth lemma of the third book of his Principles. He diftmguifhes two cafes of this problem; the firft, when the ordinates drawn from the given points to any line given in pofition, are at equal diftances from each other ; and the fecond, when thefe ordinates are not at equal diftances. He has given a folution of both cafes, but without dernonft ration in that place, which has fince been fupplied by himfclf and others. See his Methodus Differen- tiahs, publiflied with other tracts of the fame author, by Mr. Tones, London, 1711; and Stirling's explanation of the Newtonian differential method in the Philof, Tranf. N°. 362. Cotes y de Methodo differential! Newtoniana, in his works pub- liflied by Dr. Smith ; Herman. Phoronomia in Append, p. 389 ; fee alfo Le Seur and Jacquicr, in their Comment on Sir Ifaac's Principles, torn. ii. p. 42, feq.
Where it is to be obferved, that the methods there demon- ftrated by fome of thefe authors extend to the defcription of any algebraic curve through a given number of points, which Sir Ifaac, writing to Mr. Leibnitz, mentions as a problem of the greateft ufe.
By this method, fome terms of a feries being given, and fup- pofed to be placed at given intervals, any intermediate term may be found nearly ; and this therefore gives a method for interpolations. Newt. Meth. Differ, prop. v. Any curvilinear figure may alfo be fquared nearly, of which fome ordinates may be found. Newt. ibid. prop. vi. And this method may be extended to the conftruclion of ma- thematical tables by interpolation. Ibid. Hzfchol. p. 100. The fucceffive differences of the ordinates of parabolic curves, becoming ultimately equal,, and the intermediate ordinate re quired being determined according to Sir Ifaac's rules, by thefe differences of the ordinates, is the reafon of this me- thod's being called the differential method. To be a little more particular :
The firft cafe of Sir Ifaac's problem amounts to this ; a fe- ries of numbers, placed at equal intervals being given, to find any intermediate number of that feries when its interval from the firft term of the feries is given.
Subtract every term of the feries from the next following, and let the remainders be called firft differences ; then fubtract. each difference from the next following, and let thefe remain- ders be called fecond differences ; again, let each fecond dif-
Append. k
D I F
fcrence be fubtrafted from the next following, and let thefe remainders be called third differences, and fo on : then if A be the firft term of the feries, d' the firft of the firft differ- ences, d" the firft of the fecond differences, d'" the firft of the third differences, tSe. and if x be the interval between the firft term of the feries and any term fought, E, that is, let the number of terms from A to E, both inclufive, be — *+ij then will the term fought,
e = a + "S ' + ±*El r i ±EE±E~~
I ' T . 9 r T 11
2 . 3-4 riian in this
"+-7T-
d""+ &c. which feries differs from the Newto. d'" _ d""
3* '
, -, .a ,
that the quantities . —
1.2.3.4
ufed by Sir
here ufed, fignify the fame with d", d'" Ifaac.
Hence if the differences of any order become equal, that is; if any of the quantities d", d'", d"" become =0, we fhall have a finite expreffion for E, the term fought ; it being evi- dent, that the feries muft terminate when any of the differ- ences d", d'", ISc. become := o.
It is alfo evident that the coefficients -- .
ev.of
. ' 1.2 ' the differences, are the unc'ne of the binomial theorem. A method may be deduced from the foregoing expreifion, of finding the funis of the terms of fuch a feries. For if we imagine a new feries, whereof the firft term fhall be = o, the fecond = A, the third = A + B, the fourth = A + B + C, the fifth = A -f- B -f C + D, and fo on, it is plain that the affigning one term of this feries is finding the fum of all the terms A, B, C, D, (gc. Now fince thofe terms are the differences of the (urns o, A, A -f B, A+B + C, A -f-B + C-f-D, and that, by the fuppofition, fome of the dif- ferences of A, B, C, D, &c. are := o ; it follows that fome of the differences of the fums will alfo be = o ; ar.d that
whereas in the feries A -i-xd' 4- ~ d", IS c. where-
by a term was affigned, A reprefented the firft term, d' the firft of the firft differences, and that x reprefented the interval between the firft term and the laft, we are to write o inftead of A, A inftead of d', d' inftead old", d" inftead old'", EsV. and jr-fi inftead of x ; which being done, the feries
expreffing the fums will be o + i
■l.A-f-
x+ I.
d' +
x+l.x.x _^ r+ ^ or _ _
1-2-3 :
2 3
lf x.x — I . x — 2 « + — ^ T~~ r d" , c?r. Or again, if the real num- ber of terms of the lines be called z, that is, if z =*+ I, or % — i = x, we fh all have the fum of the feries =nX
, , z — r "„ . z— i.z— 2 „ % — i . z — 2 . z — -i
A + — — d' + .-t d"-\ i d"+
2 2 -3 -2.3.4 T
&c. See De Mmwe, Doflr. of Chances, p. 52, 53. Mifc. Analyt. p. 153.
For inftance, let it be required to find the fum of a feries of thefquaresofthe natural numbers 1 -1-4 + 94- 164-25+36 The:
=: 1
d'
4
3^
9
5
16
7
25
9~
36
11 "
The fum confequcntly will be,
% .— I r r v
-zXi+— -X3-f--
3ZXI + V + -
2. 3
- 3% + 2
X2
= zX
p 64-0,2; — q-\-izz — 6z-|-4
' 6^ "
i+_3j* + 2 % z z.i-\-z. i-f-22: &.7.13
-=91
6 6
This eafy example will be fufficient to fhew the applications of the rule. Thofe who are defirous of feeing its ufe in que- ftions of chance, may confult Mr. de Mdivre's Doctrine of chances, p. 53, feq. Various other inflances of the ufe of this rule in finding fums of progreffions of figurate members, £sV. may be i'ecn in Mifc. Analyt. p. 154, feq. As to the differential method, it is to be obferved, that though Sir Ifaac and others have treated it as a method of defcribing an algebraic curve, at leaft of the parabolic kind, through any number of given points; yet the confideration of curves is not at all eflential, though it may help the imagination. The defcription of a parabolic curve through given points, is the fame problem as the affigning of quantities from their given differences, which may always be done by algebra, and by G the
