But, by Ampère's result, that two closed circuits act on one another as two magnetic shells, it should be - ffdsfds (S.Uv,Uv+3 SUSU, 1). = 1 Ta Ꭲ which are consistent with one another, and which lead to and the mutual potential of two elements is of the form which is the expression employed by Helmholtz in his recent paper. (Ueber die Bewegungsgleichungen der Electricität, Crelle, 1870, p. 76.) Monday, 22d December 1873. Sir W. THOMSON, President, in the Chair. Professor Andrews, Hon. F.R.S.E., Vice-President of Queen's College, Belfast, gave an Address on Ozone. Monday, 5th January 1874. Professor Sir WILLIAM THOMSON, President, The following Communications were read: 1. A new Method of Determining the Material and Thermal Diffusivities of Fluids. By Sir William Thomson. 2. Continuants-A New Special Class of Determinants. By Thomas Muir, M.A., Assistant to the Professor of Mathematics in the University of Glasgow. 1. A determinant which has the elements lying outside the principal diagonal and the two bordering minor diagonals each equal to zero, and which has the elements of one of these minor diagonals each equal to negative unity, may be called a Continuant. is a continuant of the fourth order. 2. A continuant is evidently a function of the elements of the principal diagonal and the variable minor diagonal, and of these alone. Let this function be denoted by K. The above continuant, for example, may then be written 3. By the cyclical transposition of rows and thereafter of columns, we establish a first law of continuants, viz.:— (a (an ..... bn-1 b. b1 b2-1)= Ka a-1a a (I.) ) 4. By expansion of the continuant in terms of its principal minors we have 5. From this we see how to evaluate a continuant for special values of its elements, and also to change a continuant into the ordinary notation, i.e., to free it of determinant forms. Thus, 46897 K(723145) would be evaluated by first evaluating K (475), thence K (19475) 6. By means of Laplace's expansion-theorem we can establish a result which includes (II.) viz., and, using instead the present author's extension of Laplace's theorem, we arrive at a still more general proposition, viz., where of course h<p<n. An important particular case is that for which pn-1 and h = 2. K 7. Another result which is easily proved by induction is we may call a a,..... aa the main diagonal, and b,b,..... b-1 the minor diagonal; a, a,....., b1, b2,.... being known as elements. When each element of the minor diagonal is unity, the continuant may be called simple, and in writing such continuants we may agree to omit the minor diagonal, putting, for example, 9. If the elements of the first column of the determinant K(1 a,a,... a) be subtracted from the corresponding elements of the second column, it will be seen that thence, with the help of (III.), we can show that K(... a, b, c, 0, e, f, g,...)= K(... a,b,c+e,f,g,...) (VII.), and from this that K(... a, b, c, 0, 0, 0, e, f,...) = K(... a, b, c + e, f,...) and so, generally, when the number of consecutive zero elements is odd. 11. Similarly, from (II.) VOL. VIII. K (0, 0, α, α, ・ ・ ・ ɑ„) K (a, a,... a), ... 2 G and from this, with the help of (III.), we can prove that K(... a, b, 0, 0, e, f,...) = K(... a, b, e, f,...). (VIII.), and so, generally, when the number of consecutive zero elements is even. 12. Using the ordinary process of finding the greatest common measure of two numbers, we may establish another special property of simple continuants, viz., that, whatever a, a,... may be, is prime to K (a1, a,, . . . α-1, ɑ„) K (a,, a,... a-1), K(a,... a-1, a), K (a,-1, a,,... a), and K (a, a,... a-1). 13. When both diagonals of a continuant are the same when read backwards as when read forwards, it may be called symmetrical. In connection with simple symmetrical continuants, the following identities may be mentioned: K(a,,a,,.,,... a, a,) = K(a1, a,,...a,-1) {K (a1, u,,... a) 14. The value of the special study of this class of determinants lies in the fact that by means of them the convergents of a con |