This is, of course, an important point.

Leslie Ellis expresses

Shem was a son of Noah

by

S = 8N;

but, so far as I have yet studied the matter, I am inclined to hold thats must be considered as in general a plural symbol, and that

S = $N

asserts implicitly that Shem was the only son of Noah. The truth Shem was a son of Noah

20. Let ƒ denote father of and m denote mother of. Then

21. To find expressions for one's ancestors of the nth generation

then fff + mff + fmf + mmf + ffm + mfm + fmm + mmm . These are the expressions for one's great-grandparents.

And so on.

The maximum number of ancestors of the nth generation back which a person can have is 2", and the minimum number according to the Laws of Consanguinity is 4.

22. According to the above, the relation of great-grandmother may denote any one of the four different relations—

mff, mmf, mfm, mmm;

and, taking into account the gender of the great-grandchild, there will be eight different relationships.

23. The notation I have framed is of great use in showing the ambiguities of the common terms of relationship. Thus uncle may may mean any one of eight things, or a combination of these. For example

But that is morally, if not physiologically, impossible unless A =B,

1

σ = σ

1

Observation.-Neither o nor s can be equal to 1.

25. To express that

8

σ

A is the brother of the brother of B.

The expressions for brother are―

Subscript letters are to be understood." If in the case of 1 or 7 the subscript letters are the same, then

that is, let A be the son of a male who is the mother of the

male B.

But this is impossible in the case of the human species, where sex is monœcious. Hence A is imaginary; and therefore

27. The different permutations of the four fundamental symbols used directly and inversely may be exhibited in a table. I append one-fourth part of the complete table, marking the expressions which are impossible or which denote coincidence.

FEFE

VOL X.

8

1 1

d

8=0

1

2 c

28. The Marriage with a Deceased Wife's Sister Bill would make the following among other equations possible :

29. As the analysis of relationship is important not only in itself, but also as throwing light upon the nature of operators in Mathematics, I propose to continue the investigation, and to bring the results before the Society at a future meeting.

Monday, 2d June 1879.

SIR C. WYVILLE THOMSON, Vice-President, in the Chair.

The following Communications were read:

1. On the Carboniferous Volcanic Rocks of the Basin of the Firth of Forth: their Structure in the Field and under the Microscope. Second Paper. By Professor Geikie.

2. Additional Observations on the Fungus Disease affecting Salmon and other Fish. By A. B. Stirling, Assistant Curator of the Anatomical Museum of the University of Edinburgh.

In my former paper, read before the Society in June 1878,* I gave an account of observations which I had made on the fungus disease affecting salmon, and described the character of the fungus, which I referred to Saprolegnia ferax.

In the present communication I propose to relate additional observations, and to discuss the theories which have been advanced by different writers in explanation of the cause of the disease. Four theories have been advocated, namely-pollution of rivers, overcrowding, absence of frost, diseased kelts and addled ova.

In reference to the theory that pollution is the cause of the

*See Proceedings of that date.

fungus disease of salmon and other fish in the rivers at present affected by it, I think it is only necessary to relate the fact, that diseased fish are found in those rivers many miles above all sources of pollution, to prove that it cannot have originated from that cause. In the Eden River, for twelve miles above Carlisle, a district in which there is no big town or other sources of pollution, the fungus disease has been found as deadly as below Carlisle after the sewage of the city has entered the river. In the Tweed also, both trout and greyling, which are non-migratory fish, have been found affected with fungus where no source of pollution is known to exist; for I have obtained trout from near Broughton, and greyling from near Stobo, both of which are from seven to ten miles above Peebles, the town highest up the Tweed. Mr Buckland also, in his Seventeenth Report on the Salmon Fisheries, England and Wales, 1878, states "that we must look to other circumstances in order to diagnose the origin of the mysterious disease."

The theory of overstocking as the cause of the disease has been advocated by Mr Buckland in the same report. He considers that "owing to the absence of freshes (spates) in a river, the spawned fish do not find their way to the sea, so that they accumulate in the pools in which the disease breaks out amongst them, as gaol-fever affected the crowded prisons in former times."

In May 1874 the Tweed Commissioners constructed a small pond for experimental purposes, which measured 36 feet by 16 feet, on the side of a small stream called Carham Burn, from which a run of water was supplied to the pond by a drain pipe. On 7th May 1874, 130 sea-trout smolts, the average length of each being 8 inches, were taken from the Tweed and placed in this pond. After an interval of two years they were specially examined, weighed, and measured on the 25th May 1876. Seventy fish were found in the pond, the average length of each was 12 inches; they were now in the whitling stage, and in fine condition. After another interval of two years there was another examination, when they were weighed and measured on the 23d May 1878, when sixty-six sea-trout, of the average length of 14 inches, were found in the pond.

In the interval between the examinations, and probably in the season 1876-77, the fish had spawned in the pond, and a numerous