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It has been assumed in these rules that the titration has yielded proportional results; but these are not always obtained. There can be no doubt that in any actual re-action the proportion between any two re-agents is a fixed one, and that if we double one of these then exactly twice as much of the other will enter into the re-action; but in the working it may very well be that no re-action at all will take place until after a certain excess of one or of both of the re-agents is present. In titrating lead with a chromate of potash solution, for example, it is possible that at the end of the titration a small quantity of the lead may remain unacted on; and it is certain that a small excess of the chromate is present in the solution. So, too, in precipitating a solution of silver with a standard solution of common salt, a point is reached at which a small quantity of each remains in solution; a further addition either of silver or of salt will cause a precipitate, and a similar phenomenon has been observed in precipitating a hydrochloric acid solution of a sulphate with baric chloride. The excess of one or other of the re-agents may be large or small; or, in some cases, they may neutralise each other. Considerations like these emphasise the necessity for uniformity in the mode of working. Whether a process yields proportional results, or not, will be seen from a series of standardisings. Having obtained these, the results should be arranged as in the table, placing the quantities of metal used in the order of weight in the first column, the volumes measured in the second, and the standards calculated in the third. If the results are proportional, these standards will vary more or less, according to the delicacy of the process, but there will be no apparent order in the variation. The average of the standards should then be taken.

It acquires a yellowish tinge. As it continues to contract it grows hotter and hotter until its temperature reaches a maximum as a white star. At this point the contraction process does not stop, but the heating process does. Further contraction is now accompanied by cooling, and the star goes through its colour changes again, but this time in the inverse order. It contracts and cools to yellow and finally to red. But when it again becomes a red star it is enormously denser and smaller than when it began as a red star. Consequently the red stars are divided into two classes called, appropriately, Giants and Dwarfs. This theory, which we owe to an American astronomer, H. N. Russell, has been successful in explaining a variety of phenomena, and there is consequently good reason to suppose it to be true. But the question as to how the red giant stars were formed has received less satisfactory and precise answers.

The principle of association, as every one knows, involves that whenever two things have been associated sufficiently together, the suggestion of one of them to the mind shall immediately raise a suggestion of the other. It is in virtue of this principle that language, as we so call it, exists at all, for the essence of language consists, as I have said perhaps already too often, in the fixity with which certain ideas are invariably connected with certain symbols. But this being so, it is hard to see how we can deny that the lower animals possess the germs of a highly rude and unspecialised, but still true language, unless we also deny that they have any ideas at all; and this I gather is what Professor Max Muller in a quiet way rather wishes to do. Thus he says, "It is easy enough to show that animals communicate, but this is a fact which has never been doubted. Dogs who growl and bark leave no doubt in the minds of other dogs or cats, or even of man, of what they mean, but growling and barking are not language, nor do they even contain the elements of language."

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