It may occur to the reader that we ought also to distinguish Qualitative and Quantitative Variations as two orders of phenomena to which the present method is applicable. But, in fact, Qualitative Variations may be adequately dealt with by the foregoing methods of Agreement, Double Agreement, and Difference; because a change of quality or property entirely gets rid of the former phase of that quality, or substitutes one for another; as when the ptarmigan changes from brown to white in winter, or as when a stag grows and sheds its antlers with the course of the seasons. The peculiar use of the method of Variations, however, is to formulate the conditions of proof in respect of those causes or effects which cannot be entirely got rid of, but can be obtained only in greater or less amount; and such phenomena are or course, quantitative. Even when there are two parallel series of phenomena the one quantitative and the other qualitative--like the rate of air-vibration and the pitch of sound, or the rate of ether-vibration and the colour-series of the spectrum--the method of Variations is not applicable. For (1) two such series cannot be said to vary together, since the qualitative variations are heterogeneous: 512: 576 is a definite ratio; but the corresponding notes, C, D, in the treble clef, present only a difference. Hence (2) the correspondence of each note with each number is a distinct fact. Each octave even is a distinct fact; there is a difference between C 64 and C 128 that could never have been anticipated without the appropriate experience. There is, therefore, no such law of these parallel series as there is for temperature and change of volume (say) in mercury. Similar remarks apply to the physical and sensitive light-series. We may illustrate the two cases of the method thus (putting a dash against any letter, A' or _p_', to signify an increase or decrease of the phenomenon the letter stands for): Agreement in Variations (other changes being admissible)-- A B C A' D E A'' F G _p q r_ _p' s t_ _p'' u v_ Here the accompanying phenomena (_B C q r, D E s t, F G u v_) change from time to time, and the one thing in which the instances agree throughout is that any increase of A (A' or A'') is followed or accompanied by an increase of _p (p' or p'')_: whence it is argued that A is the cause of _p_, according to Prop. III. (a) (ch. xv. § 7). Still, it is supposable that, in the second instance, D or E may be the cause of the increment of _p_; and that, in the third instance, F or G may be its cause: though the probability of such vicarious causation decreases rapidly with the increase of instances in which A and _p_ vary together. And, since an actual investigation of this type must rely on observation, it is further possible that some undiscovered cause, X, is the real determinant of both A and _p_ and of their concomitant variations.