VARIOUS USES OF FACTORS 49 ascertained by factor-analysis the hypothetical correlations of g with m^ and m^ we should estimate z's most probable marks in two steps by regression equations as follows : The result is reversible ; for, by analogous regression equations, we could estimate f s probable measurement in the first test from his mark in the second, viz. fWi* = ris rzg'Vhi* The correlation coefficient being the geometric mean of the two regressions, we obtain *ia = rlg r2g or (with the alternative notation) =/u ./21. 77>£ Product Theorem. — This last equation yields what I have termed the ' product theorem/ It may be regarded as the analogue of the multiplicative theorem in simple probability.1 It is, as we shall discover in a moment, the 1 Both have close analogies to the logical multiplication of classes, rela- tions, and propositions. Indeed, I am tempted to say that the product theorems mentioned in the text are but particular cases of the more general product theorem that forms the basis of all deductive logic (if '£a& and R^c denote given relations between a and b and between I and c respectively, then the relation between a and c will be defined by Rac = Rab X Rbc ; if d and B be two classes, defined by (x) and \j/(x) respectively, then their com- mon part C will be defined by (#) X \fr(x) — where in either case the multi- plication symbol is defined in a more general way than is usual in finite arith- metical multiplication, but will include this as a particular case). Since I shall presently argue that the mathematical reasoning of the factorist, like all mathematical reasoning, is but a special example of formal reasoning generally, whether quantitative or non-quantitative, it may be worth while to exhibit the analogies more explicitly at this stage. M (i) Let us use the proper fraction -~- to symbolize such propositions as o " Some (or all) children who succeed in this test (S) are also intelligent (M)." We may then write the syllogism set out in I (a) above as follows : Number of children who are also intelligent Number of children who also gain scholarships Number of children who succeed in the test Number of children who are intelligent — Number of children who also gain scholarships ~~ Number of children who succeed in the test " If (as is assumed in paragraph I in the text) the first premiss is universal, i.e. if all the children who succeed in the test are also intelligent, the value of