third tree, and the fourth tree of fig. 3. To derive fig. 3 (bis) from fig. 3, we must fill up the trees of fig. 3 with U at the root and R, Q, P at the other knots in every possible manner, subject only to the restriction, that, reckoning up from the extremity of a free branch to the root, there must not be any transposition in the order of the symbols RQP, and taking care to admit only distinct trees. Thus the first tree of fig. 3 might be filled up in six ways; but the trees so obtained are considered as one and the same tree, and we have only the first tree of fig. 3 (bis). Again, on account of the restriction, the fourth tree of fig. 3 can be filled up in one way only, and we have thus the sixth tree of fig. 3 (bis). And thus, in general, each figure of the second set can be formed at once from the corresponding figure of the first set; or when the first set of figures is given, the expression for YX..QPU can be formed directly without the assistance of the expression for the preceding symbol X...QPU; the number of terms for the nth figure of the second set is obviously 1.2.3... n, and consequently it is only necessary to count the terms in order to ascertain that no admissible mode of filling up has been omitted.

...

The number of parts in any one of the figures of the first set is much smaller than the number of parts in the corresponding figure of the second set; and the law for the number of parts, i. e. for the number A, of the trees with n branches, is a very singular one. To obtain this law, we must consider how the trees with n branches can be formed by means of those of a smaller number of branches. A tree with n branches has either a single main branch, or else two main branches, three main branches, &c. to n main branches. If the tree has one main branch, it can only be formed by adding on to this main branch a tree with n-1 branches, i. e. A, contains a part An- If the tree has two main branches, then p+q being a partition of n-2, the tree can be formed by adding on to one main branch a tree of p branches, and to the other main branch a tree of q branches; the number of trees so obtained is A, A,: this, however, assumes that the parts p and q are unequal; if they are equal, it is easy to see that the number of trees is only A,(A,+1). Hence p+q being any partition of n-2, A, contains the part A,A, if p and q are unequal, and the part A,(A,+1) if p and q are equal. In like manner, considering the trees with three main branches, then if p+q+r is any partition of n-3, A, contains the part A,AA, if p, q, are unequal; but if two of these numbers, e. g. p and q, are equal, then the part A,(A,+1)A‚; and if p, q, r are all equal, then the part A,(A,+1)(A„+2) ; and so on, until lastly we have a single tree with n main branches, or A, contains the part unity. A little consideration will show that the preceding rule for the formation of the number A. is

completely expressed by the equation

(1—¢)-1(1—z4)-4h(1—28)-4(1—4)ཡ╚ཏྭཱ...

=1+A1x+A ̧x2 + Ag23 + A4x4 + &c.,

and consequently that we may, by means of this equation, calculate successively for the different values of n the number A, of the trees with n branches. The calculation may be effected very easily as follows:

A 1 1 2 4 9 (20) 28 47 103 178 226 (1-6)-20

A,= 1 1 2 4 9 20 48 (115) 191 354 598 (1-x8)—11

Ag= 1

115 115 330

1 2 4 9 20 48 115 (306) 469 928 (1-9)-30
306 306

A, 1 1 2 4 9 20 48 115 306 (775) 1234 (1-210)-775

775

1 2 49 20 48 115 306 775 (2009)

A,=

T

for r

1 234 5 6

7 8 9 10

I have had occasion, for another purpose, to consider the question of finding the number of trees with a given number of free branches, bifurcations at least. Thus, when the number of free branches is three, the trees of the form in question are those in the annexed figure, and the number is therefore, two, It is not difficult to see l, 25 that we have in this case (B, being the number of such trees with free branches),

1

o

rs,

I may mention, in conclusion, that I was led to the consideration of the foregoing theory of trees by Professor Sylvester's researches on the change of the independent variables in the dif ferential calculus.

2 Stone Buildings. January 2, 1856.

(1

XXIX. Remarks on a passage in the President's Address delivered at the Anniversary Meeting of the Geological Society of London on the 15th of February, 1856. By the Rev. Prof. SEDGWICK, M.A., F.R.S. &c.

To the Editors of the Philosophical Magazine and Journal. »b GENTLEMEN,

THE HE remarks which follow might be thought unworthy of publication were they not written in defence of a paper which appeared in your Journal for October and November 1854. Mr. Hamilton, in his Anniversary Address, has attacked the conclusions I vindicated in that paper. As, however, he appears neither to have learnt, by a study in the field, what are the great successive physical groups in the older divisions of British paleozoic rocks, nor to have more than a partial and one-sided knowledge of their history, I should not perhaps have been called upon to notice his statements, so far as they are purely geological, either in the way of praise or blame. But he has at the same time charged me with having irregularly gained possession of that which was the property of the Geological

Society, and "with having [in your Journal for October and November 1854] published what I must have known that I had no right to publish." These are grave charges, and in my interpretation of them, they involve nothing less than a severe censure on my moral character. For unfair dealing is on all occasions mean and disgraceful; and never more disgraceful than when it is done against the known regulations of a public body.

There is not perhaps a single geologist of much note in Europe or America who has not read Mr. Hamilton's Anniversary Address; and if there be charges in it which I think unjust and injurious to myself, I am bound to meet them by a public counter statement. It may indeed appear strange that my defence, so far as it is personal, has been so long delayed; but the delay may be readily explained. It was near the end of the month of April that I first had a hint (from my friend Dr. Fitton) of certain charges that had been brought against me by Mr. Hamilton; but it was not till several months afterwards that I obtained a copy of his Anniversary Address. When I received it I was away from the University, and without access to a single memorandum that might help to bring before my memory the facts which led to the publication of my paper in your Journal; and the meetings of the Geological Society had been closed for the summer recess. I was on both accounts literally compelled to postpone my defence; but I resolved to make it, if possible, at the first autumnal meeting after the Society was re-assembled.

In the latter part of the month of October I returned to Cambridge, and soon afterwards I had a short correspondence with Mr. Hamilton and Mr. T. R. Jones, which enabled me to correct some mistakes into which I had fallen in the first sketch of my defence; but on the 10th of November I sent my letter of vindication, addressed to the President and Council of the Geological Society, with a request "that as a matter of common justice it might be read at one of their early meetings." A very few days afterwards commenced what I perhaps might call a second correspondence between Mr. Hamilton and myself, which ended by a proposition on his part (dated November 20), that I should withdraw my letter to the Council, on the condition that he should make a public statement to the effect "that he willingly acquitted me of any intention of infringing the rules of the Society," and "that he regretted the statement he had made under a wrong view of my motives." I readily accepted this condition, and with the consent of the President my letter to the Council was withdrawn.

After waiting with no small anxiety for the public explanation which might help to set me right with the Geological Society, I

wrote more than once, both to Mr. Hamilton and to Mr. T. R. Jones, that I might know whether any explanation had been offered; and at length I was informed (in a letter from Mr. Hamilton, dated December 27) " that on the 7th of January a statement would be made by himself before the Geological Society entirely exonerating me from all charge of wilful disobedience to their laws and customs*."

While accepting Mr. Hamilton's condition, I distinctly reserved to myself the full liberty of publishing any further vindication of myself, or of my previous memoirs, which I might think right or expedient. My leading object, in what is above stated, has been to prove that I have neither shown a disgraceful indifference to a charge so publicly brought against me, nor have made any needless delay in defending myself from what I thought its injustice.

I have the honour to be, Gentlemen,
Your faithful Servant,

Norwich, January 9, 1857.

ADAM SEDGWICK.

The passage in the Anniversary Address upon which I wish to comment is word for word as follows:-"I must also correet Prof. Sedgwick's memory when he says that the Council refused to publish his next paper in 1853. It is true he makes no complaint. He admits that part of it was in a controversial form. But I must remind Prof. Sedgwick that it was only the conclusion, viz. the controversial portion, which the Council objected to publish; the body of the paper would have been duly printed; and when I wrote to inform him of this decision, and to request his sanction to the suppression of the latter portion, the reply which I received was to the effect that he could give no answer until he had seen the paper again, and judged of the effect of the intended omission. I directed the paper to be forwarded to him, and after waiting many months for a reply, the only intimation I received of his intention was finding it printed in another journal. Such a proceeding was in the highest degree irregular. The paper was the property of the Society, and Professor Sedgwick, an old President of the Society, must have known that he had no right to make such use of it without having first obtained the sanction of the Council to its withdrawal."

Reply to the above Extract.

In the first sentence of this extract the author blunders in the date; and I only pause to remark, that to stumble on the

Since this letter to the Editors was written, I have learnt that Mr. Hamilton's statement was duly made, and so far I am grateful to him for it.