Now, according to the law of probabilities alone, the chances of a person having small-pox twice will be the square of this fraction, or _one-hundredth_: so that on the average only one person in 100 would have small-pox twice if it were a matter of pure chance, and if nothing interfered with that chance. But there are interferences which modify the result. (1) Those that die of the first attack cannot possibly have it a second time. (2) It is most frequent in the very young, so that the chances of having it later in life are not equal. (3) It is an especially _epidemic_ disease, only occurring at considerable intervals, which reduces the chances of infection to those who have had it once. (4) It is probable that most persons are only liable to infection at certain periods of life, having passed which without infection they never take the disease. It seems probable, therefore, that these several conditions would greatly diminish the chances in the case of any person who had once had small-pox, so that perhaps, under the actual state of things, chance alone would only lead to one person in two hundred having the disease a second time. The above is only an illustration of the principle. Professor Vogt goes more fully into the question, and arrives at the conclusion that out of every 1,000 cases of small-pox the _probability_ is that ten will be second attacks. Then by getting together all the European observations as to the actual number of second attacks during various epidemics, the average is found to amount to sixteen in 1,000 cases, showing a considerable surplus beyond the number due to probability. Further, the proportion of deaths to attacks has from early times been observed to be high for second attacks; and it has also been observed by many eminent physicians, whose statements are given, that second attacks are more common in the case of persons whose first attacks were very severe, which is exactly the reverse of what we should expect if the first attack really conferred any degree of immunity. Now the whole theory of protection by vaccination rests upon the _assumption_ that a previous attack of the disease is a protection; and Professor Vogt concludes his very interesting discussion by the remark: “All this justifies our maintaining that the theory of immunity by a previous attack of small-pox, whether the natural disease or produced artificially, must be relegated to the realm of fiction.” If this is the case, the supposed _probability_ or _reasonableness_ of an analogous disease, vaccinia, producing immunity wholly vanishes.