. On the Theory of Consistence of Logical Class-Frequencies, and Its Geometrical Representation. ll less than 0*5, but that we thensubstitute (1—^3) for pg so as to make one ratio greater than 0*5. How is theoriginal surface altered ? Substituting {l—p)^ fo^^Ps amounts to substituting y for if x^, y^, z^ be the original co-ordinates, x^, 7/^, Zc^ the co-ordinates after thesubstitution, we must have OF LOGICAL CLASS-FREQUENCIES, ETC. 117 «A^ O —— tAJ ?] V% ^ P\ - Vi Zc 2 Ih z 15 since (Ay) ^ (A) ^ (AC). (By) = (B) ^ (BC). The first surface is therefore changed into the second by a simple
. On the Theory of Consistence of Logical Class-Frequencies, and Its Geometrical Representation. ll less than 0*5, but that we thensubstitute (1—^3) for pg so as to make one ratio greater than 0*5. How is theoriginal surface altered ? Substituting {l—p)^ fo^^Ps amounts to substituting y for if x^, y^, z^ be the original co-ordinates, x^, 7/^, Zc^ the co-ordinates after thesubstitution, we must have OF LOGICAL CLASS-FREQUENCIES, ETC. 117 «A^ O —— tAJ ?] V% ^ P\ - Vi Zc 2 Ih z 15 since (Ay) ^ (A) ^ (AC). (By) = (B) ^ (BC). The first surface is therefore changed into the second by a simple transformation ofco-ordinates, ^ if we are dealing with an actual model of the surface, by turning itover and shifting it. Fia*. 5 is drawn to illustrate the nature of an actual transformation. It is drawnfor the values Pj = 0-35 J9^ = 0*4 _p3 = 055, substituting (1 — 0*45) for the 0*45 assigned to pg in fig. 4a, p. 115. The two figuresare similarly lettered. The model of fig. 4 has been turned over, round an axisparallel to the axis of x, through a half revolution. 5/ 0*3-. g. 5. § 23. If (1—Ps), {^—p%), (l-~Pi) be successively substituted for 2%, 2^3, pi the trans-formations are as follows :— (1.) Substituting (1—pg) for pg z^ x^ —— JU\ y% -Pi h -Ps 18 MR. Cx. UDNY YULE ON THE THEOEY OJ^^ CONSISTENCE (2.) Substituting (I—P2) f^i^2 JUo —— (3.) Substituting {1—Pi) forpj^ Pl , -^p^ . — a; 2/a ^ ^^j . - !/. 1 - -P3 - - ^3 — 1 - -Pa Ps + ^1 X, !/4 2i 4 1 7 Pz Ju<) 0 1 - Pi. ~ Pa + ,«! Pi - ?/3 - 1 ^^ PI »3 + ?/] ^Z 1 - Pa - P3 + ^1- 7 The second and third cases are obtained, like the first, by simply expanding. Thus— {I3y)/{u), and (/3y) :== (y) -^ (By) =. (U) ^^ (C) ^ (By) ==: (U) ^ (B) - (G) + (BC) or, dividing by (IT) ^3 ^ ^ - Ps 2;c s 1 ._ p^ ..^. p^ J^ Zy as a Dove. § 24. The correctness of the transformations given may, of course,directly« Thus suppose Pl <P^ < Ih < 0^5 then the equations to the bounding planes of
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