of the nerve-area. Externally are circular coat, hypoderm, and cuticle. Posteriorly the cords are connate, and a single neural canal of considerable size exists superiorly.

SERPULIDE.-In Protula protensa the cords anteriorly are situated under the great longitudinal dorsal muscles as widely separated trunks. They then fall into position and approach the increasing longitudinal ventral muscles-a large neural canal being at the inner border. Posteriorly the neural canal is proportionally larger than in front, but the relations of the nerves are similar. At their commencement in front the nerve-cords are internal, and have beneath them all the tissues under the dorsal muscles. After reaching the inner border of the ventral longitudinal muscles they have externally (i.e., inferiorly) a thin layer of longitudinal muscular fibres, the circular coat, hypoderm, and cuticle. A transverse band bounds them superiorly. Posteriorly the first-mentioned (longitudinal) layer is absent externally. In Serpula vermicularis, L., a similar arrangement occurs, but the neural canal is external, that is, next the ventral longitudinal muscle, and the thin longitudinal stratum is internal.

4. On the Application of Graphic Methods to the Determination of the Efficiency of Machinery. By Professor Fleeming Jenkin.

(Abstract.)

The general scope of the paper was to show how by graphic methods we might find the relation between effort exerted at one part of a machine and the resistance overcome at another part. It was shown that for a given machine at any instant a linear frame might be substituted, such that the stresses in the links corresponded to the pressures at the joints between the elements of the machine, including the pressures due to driving effort and resistance. Numerous examples were given of the application of this frame, called the dynamic frame, to the solution of the above problem, the friction, inertia, and weight of all the parts being taken rigidly into account.

5. On Professor Tait's Problem of Arrangement. By Thomas Muir, M.A.

The problem in question is-To find the number of possible arrangements of a set of n things, subject to the conditions that the first is not to be in the last or first place, the second not in the first or second place, the third not in the second or third place, and so on.

A little consideration serves to show that we may with advantage shift the ground of the problem to the theory of determinants. For the sake of definiteness take the case of five things, A, B, C, D, E. Here A may be in the second, third, or fourth places only; B in the third, fourth, or fifth places only; and similarly of the others a result which we may tabulate thus:—

an A being written in the places which it is possible for A to occupy, and a dot signifying that the letter found in the same line with it may not occupy its place. Hence, to obtain the various arrangements, we see that for the first place we may have any letter that is in the first column; for the second place any letter that is in the second column, provided it be not in the same line with the letter taken from the first column; for the third place, any letter that is in the third column, provided it be not in the same line with either of the letters previously taken, and so This law of formation, however, is identical with that in accordance with which the terms of a determinant are got from the elements of the matrix; so that the problem we are concerned with is transformed into this: Find the number of terms of the determinant of the nth order of the form,

on.

where the dots and asterisks denote zero elements and non-zero elements respectively.

In the solution of this, determinants of a set of other forms require to be considered, viz.,

*

&c.

in all of which the elements of the main diagonal are zero; the elements of the adjacent minor diagonal being in the first also all zero, in the second all zero except the first element, in the third all zero except the second element, and so on. Let us denote the number of terms in these when they are of the nth order by

X(n), X1(n), X2(n), . . . .

and let the number of terms sought be

Y(n).

Further, let any determinant-form with dots and asterisks stand for the number of terms in such a determinant; the four forms above, for example, being thus symbols equivalent to ¥(5), X。(5), X(5), X(5), respectively.

VOL. IX.

3 E

Beginning with the first of the x forms we see it to be transformable into

and the second term of this may in like manner be changed into

and the second term of this again into

*

the second term of which is zero.

Hence we have the result,

Xo(n) =¥(n) + ¥(n − 1) + Y(n − 2) + · +¥(3), .

...

Of the other forms it is clear to begin with that

and treating the distinct cases as we have treated X., we equally readily see that

which, on eliminating X, X, ... from the right-hand members, become

X1(n) = xo(n)+x。(n − 1)+x(n − 2)

X(n) = x(n)+x(n − 1)+x(n − 2) +x (n − 3)+x(n − 4)

X3(n) = x(n)+x(n − 1) +x (n − 2) + x (n − 3) + x (n − 4)+x(n − 5) +xo (n = 6)

Xn−2(n) = Xo(n) +x (n − 1) + x(n − 2) +x (n − 3)+x (n − 4)

Xn-1(n) = X(n)+x(n − 1)+x(n-2).

Here the second of the series of right-hand members has two terms more than the first, the third two terms more than the second, and so on until we approach the middle of the series, when, if n be odd, the two middle right-hand members are found to be the same as the one preceding or the one following them, the whole four ending thus

and if n be even, the one middle right-hand member is found to be greater by unity than the one preceding or the one following it, and to end thus

...+xo(5)+xo(4)+x。(3)+1,

the 1 arising from the fact that the above process of reduction, in the case of X(n), leads us finally, not to