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Under the sixth proposition he considers the practicability of obtaining in North America, from sea water, a supply of bay salt adequate to its consumption. From the comparative situation of France and North America; from the natural and artificial advantages of the latter, from the intensity of its heat, from the early maturity of its fruits; from all these considerations the necessary inference, as he conceives, is, that by inconsiderable exertions, and at a moderate expense, the inhabitants of that portion of the globe might prepare salt in such abundance, as not only to satisfy the demand of their fisheries, but even render their commerce more flourishing by its exportation.

(To be continued.)

ARTICLE II.

Explanation of the Characteristics d and d. By Mr. Adams.

SIR,

(To Dr. Thomson.)

Stonehouse, near Plymouth, July 6, 1817. BEING of opinion that the following explanation of the characteristics d and d, together with the general formula derived therefrom, may be useful to youths just entering upon the differential and integral calculus, your inserting them in the Annals of Philosophy will much oblige,

Sir, your most obedient servant,

JAMES ADAMS.

On the Characteristics d and d.

Since by differentiating with the characteristic d, we have

d (x) = d x

d (x2) = 2 x dx

d (x3) = 3 x2 d x

1

d (x") = n x"1 d x

-2

d2 (x") = n (n − 1) x2 -2 d x2

&c. &c. dx being constant.

So by differentiating with the characteristic d, we have

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From whence, if V be a function of y, we may easily arrive at this fundamental theorem,

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Otherwise thus: since the variation of any quantity depends upon that quantity, it is evident that the variation of any primitive must be a function of that primitive.

Therefore dy = & Y

And dy'

= function of y

= y = function of y

Hence dy - dy = q y′ — øy = do y = ddy.

But dyyy.. ddy = dy dy

Consequently d dy ddy.

Hence it follows,

=

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(See Translation of Lacroix' Differential and Integral Calculus,

page 439.)

The Same from the Consideration of Curves.

Let A M, B N, be any two curves; and PM, Rr, two consecutive ordinates; the other lines as per figure.

Put Rvy, then vr = dy = variation of the ordinate R v, and N n = differential of the same ordinate.

In like manner, MN will represent the variation of P N, and M m the A differential of Rr.

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BR

P

Hence v r+ M m = M N + Nn; that is,
+ d y) + dy; or,
dy + ddy + dy :. d d y = d d y.

dy + d (y +dy) = dy + dy + d dy =

d (y

In any two curves.

COROLLARY

ddy.

The difference of the variations of two consecutive ordinates is equal to the difference of their differentials.

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For vr+ M m = M N + N n .. v r MN M m

N n.

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The expression "S♪V = d'ƒ V, may be proved as follows
Let w = "SV = nth integral of V.

Then d" w = V .. d d" V = d V = ddw (by the foregoing)
Therefore V = "f d" dw = d"-" dw = dw = d "S V
That is, "V = ♪ "SV..

(B).

Suppose and to represent two variable functions; and let it be required to find the variation of " de. From the theorems marked (A) and (B), we have ♪ ("S & dm 0) = "ƒd (ø £m 0) = "S (p = dm 0 + ♪ & dm 0), Φ

dm

And dTM (0) = & dm do + dm & d 0 = q d dTM 0 + dm & do,

Therefore, ddm 0 = dm (pdf)
Φ

Then, by substitution,

d("S¢ dTM 0) = "S {dTM (ø do) —

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φ

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dm

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d

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d (Sq d 0) = q d0 + S (ò ø d 0 — d qd0).

ARTICLE III.

Solution of the Equation 4 x = x. By Mr. Horner.

THE general method of effecting this solution when a particular value fofis known, has been given by Mr. Babbage in his Essay on the Calculus of Functions. (Phil. Trans. 1815.) He has favoured us with some additional illustrations of the subject in the second volume of the Journal of Science; where he has employed a very ingenious artifice to solve the particular cases 42 x = x, and 4+ x = x.

The solution of the general equation on the same principle would have led Mr. B. into details inconsistent perhaps with the breadth of plan which the developement of a new calculus requires; and which he might have felt himself the less disposed to pursue, in consequence of thinking it "probable that this solution, although a very extensive one, does not contain all the possible answers; and if we have regard to the utmost generality, the solution [of 4a I 9] must be deduced from that of the equation

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I cite these words from Mr. B.'s concluding observations, in the Journal just mentioned, on the solution of 44 x = x, which is contained in the formula

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And, with all the deference due to this gentleman's profound science, I cannot help attributing his hesitancy in this instance to his having overlooked a principle which he has so happily applied to functions of the first order in Prob. VIII. of his Essay. It is by no means restricted to such expressions, but adapts itself with the evidence of an axiom to pure functional equations of all orders. I refer to the principle that the arbitrary constants in such equations may be exchanged for any symmetrical functions of all the inferior functions of the variable concerned. In the formula just quoted, for example, we are at liberty to make

a = a {x, 4 x, 42 x, 43 x}

1 = ß {x, 4 x, 42 x, 43 x}

C = y {x, 4 x, 42 x, 43 x}

And when these substitutions are made, nothing-as it appears to me-can be added to the generality of the solution.

By this method, it is true, we arrive frequently at implicit func

tions; but this circumstance does not detract from the perfection of the solution, since the results are placed within the reach of known processes.

The proof and illustration of these remarks will be attended with no difficulty, as far as they regard the simple case

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a symmetrical function of x and y, which can be readily identified, by means of the arbitrary constants, with any proposed function similar to

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and, therefore, à fortiori, with any particular solution of the case. EXPER. I.-When ab 1, c = 0, and 4 x = x2, we have

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of which again the algebraic reduction produces

y = ÷ {e x + √ a + (e2 4) x2}

I have selected these examples, because their final results are the only ones given by Mr. B. whose identity with the general formula is not self-evident.

Satisfied with a high degree of probability that a similar solution will be also universal in regard of the superior orders, I shall proceed to the general equation

4" x = x.

a b x
c d x

Mr. B. has remarked, that if fx =++, fx will be =

A + B x
C + D x

where A, B, C, D, are given functions of a, b, c, d; "and these latter quantities may always be so assumed, that A = O, BC, and D O." Nothing more appears to have been intended by the last observation than what is evident from the elementary principles of elimination, that because we know the values of four independent functions of a, b, c, d, we can determine the values of these quantities. But in entering into the calculations, we discover the curious fact, that the conditions here recited

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