r>>jjCv
in <Ae theory of value and prices.
81
By passing sections successively through the point I, we may nar-row the discussion to as few variables as we choose. We may thusselect any three and discuss them as before in real space (cf. § 4).
§ 9. Analytical.
For those familiar with multiple algebra, that is with the quater-nion analysis of Hamilton, the “ ausdelmungslehre ” of Grassman, orthe vector analysis of Prof. J. Willard Gibbs , the foregoing geo-metrical simplification will lead to a striking analytical simplifica-tion.*
Let I, II, . . . N, be vectors to the points I, II, ... N from theorigin. Let TJ„ U,, etc., represent the total utility at the pointsI II, etc. Let vU,, vll,, etc., be vectors to represent in magnitudeand direction the maximum rate of increase of utility at the pointsI, II, etc. (i. e. in the “maximum directions”).
The conditions of equilibrium expressed in § 7 become :
(1) vU, = F(I); V U, = F(II) ; . . . vU„ = F(N)
(2) V U. 00 V U, OO V U, 00 ... 00 V U„
(3) I + II + III + . . . + N = 0
(4) I. vU, = II. vU,= . . . = N . vU „=0
The first equation represents the several utility distributions.The second means that the “ maximum directions ” are alike ; thethird that the amount of each commodity produced and consumedcancel, and the fourth that for each individual the values of produc-tion and consumption cancel.f
* See J. W. Gibbs’ Vector Analysis, p. 16, § 50.
f The scalar equations which the preceding vector equations replace canreadily be deduced from them. Let a, b, c, etc., be unit vectors along theA. B, C, etc. axes. Multiply vTJi=F(I) by a, b, c, etc. respectively. We ob-tain m equations of the form vUi . a=F(I) .a or:
= F(A„ B„ Ci, ... . M,).
Likewise m scalar equations are contained invU, = F(I1), etc.
Again from (2) since vUi oo vU s ,
VU, . a : vUi . 6 = vUj . a : vU, . tor:tfU dU _ dU dTJdAi dB t dAj ' dB 3 .
. . M,. Likewise for vU a , etc.a + II . a 4- m . a + . . . + N . a = 0 orA, + Aj -f- A, -t- . . . + A„ = 0.
. M, making m equations.
Likewise for G\. D,, .Again (3) yields I
Likewise for B, C,
Trans. Conn. Acad., Vol. IX.
6
July, 1892.