in the theory of value and prices. 81 



By passing sections successively through the point T, we may nar- 

 row the discussion to as few variables as we choose. We may thus 

 select 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 anal3''sis of Hamilton, the " ausdehnungslehre " of Grassman, or 

 the 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, . . . IsT from the 

 origin. Let U^, TJ^, etc., represent the total utility at the points 

 I, II, etc. Let V Uj, v U^, etc., be vectors to represent in magnitude 

 and direction the maximum rate of increase of utility at the points 

 I, II, etc. (i. e. in the "maximum directions"). 



The conditions of equilibrium expressed in § Y become : 



(1) VU, ^F(I); vU,=rF(II); . . . vL\ = F(N) 



\^2) vUj a vU^oc vUg a ... oc V U^ 



(3) I -fll + III + . . . + N = 



(4) I . vU, =: II . vF^ = . . . = N . vU„ = 



The first equation represents the several utility distributions. 

 The second means that the " maximum directions " are alike ; the 

 third that the amount of each commodity produced and consumed 

 cancel, 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. 



\ The scalar equations which the preceding vector equations replace can 

 readily be deduced from them. Let a, ?>, c, etc., be unit vectors along the 

 A, B, C, etc. axes. Multiply vUi=F(I) by a, &, c, etc. respectively. We ob- 

 tain m equations of the form vUi . a = F(I) . a or: 



^ = F(A„ B., C\, .... MO. 

 aAi 



Likewise ^n scalar equations are contained in v Ua = F(ll), etc. 

 Again from (2) since vUi a vUa, 



vUi . « : vUi . ?> = vU-i . rr : v U2 . ^> or: 

 d\5 _ r/U _ r/U _ r/U 

 dAi " dBi dKi ' dBy. 

 Likewise for Ci, D,, . . . Mi. Likewise for vUa, etc. 



Again (3) yields I . a + ll . a + lll . a+ . . . + N . </ = or 

 A, +A0 + A3+ . . . +A„ = 0. 

 Likewise for B, C, . . . M, making in equations. 



Trans. Conn. Acad., Vol. TX. 6 July, K^i»-J. 



