The special function of Concomitant Variations seems to be to establish the exact quantitative relation between the varying cause and the varying effect. To illustrate: As a general law it is known that bodies attract each other in varying degrees according to their distances apart and according to their relative sizes; by Concomitant Variations this law has been given definite quantitative value and reads like this: “Bodies attract each other directly as the product of their masses, and inversely as the square of the distance between them.” This illustration suggests that the variation between antecedent and consequent may be _direct_ or _inverse_. The error most common in this method is the assumption that the quantitative relation between two varying phenomena will always be according to a _constant ratio_. For example, when being reduced from a high temperature to 39⅕° F., water steadily contracts; but at 39⅕° F. it commences to expand until it becomes ice. Thus the ratio of contraction of water is constant only within certain limits. In any event the established ratio of variation can with absolute safety be applied only to the _instances investigated_. Another disadvantage incident to this method, is the situation of two elements varying together constantly, and yet having no causal connection whatever. =6. THE METHOD OF RESIDUES.= (1) Principle stated. As stated by Mill the principle of residue is this: “Subtract from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents.” In simpler form the notion is this: _Subtract from any phenomenon those parts of it which are known to be the effect of certain antecedents, and what is left of the phenomenon is the effect of the remaining antecedents_. (2) Principle symbolized. _Antecedent_ _Consequent_ A x B y C z The total cause of the phenomenon xyz is ABC. But it is known that the cause of x is the antecedent A; whereas the cause of y is the antecedent B; hence it is concluded that the cause of z is the antecedent C. (3) Concrete illustrations. _Problem_: To find the weight of coal. _Antecedents_ _Consequents_ Weight of driver, } Weight of wagon, } = 4200 lbs. Weight of coal. } Weight of driver, } = { 200 lbs. } = 2200 lbs. Weight of wagon, } { 2000 lbs. } Hence we may conclude that the weight of coal is 4200 lbs. − 2200 lbs., or 2000 lbs.