produce in the mind a crystallisation of ignorance; and it is to prevent this that I have so persistently kept before the student the gramme, the dyne, the erg, &c. Hence in this Chapter Poisson's equation always appears in the form V2V=-4πуp, in which y is the C. G.S. constant 1 dyne of gravitation-viz. about and a familiarity 1543 x 104 with its value gives the student a useful idea with regard to the nature of gravitation. In this Chapter I have also ventured to introduce the term 'Laplacian' with reference to those remarkable coefficients which occur in the development of the reciprocal of the distance between two points. The general term 'Spherical Harmonic' is, of course, retained; but it seems to me that the name of Laplace ought to be explicitly connected with the branch of Mathematical Physics which he did so much to develop, and which has now become of such great importance. The pure mathematicians having their 'Jacobians,' Hessians,' 'Cayleyans,' &c., the term 'Laplacian' is surely justified. R. I. E. COLLEGE, COOPER'S HILL, GEORGE M. MINCHIN. A new edition of the Second Volume was called for in the summer of 1888. It is to a very great extent a reprint of the previous edition; but it treats much more fully of Conical Angles, contains new Articles on Line- and Surface-Integrals and Magnetic Shells; and, finally, an improvement in the method of treating some questions of Strain and Stress, for which I am indebted to Professor Williamson, F.T.C.D. It is satisfactory to know that the introduction of the gravitation constant has met with high approval, and has found explicit recognition in some recent papers by able writers. COOPER'S HILL, October, 1888. G. M. M. In the example in p. 115 the value of G should be 376 kilogramme-decimètres. Alter the result accordingly. The following corrections in Vol. II. were kindly communicated by Professor Hoover, Ohio University, after the sheets had gone to press : P. 271, in Ex. 12, omit the term 1 within the brackets { }. P. 281, line 19, for sin 0 read cos 0. P. 298, line 3, for sec P'PO read sec P'QO. P. 299, line 25, for C read C'. P. 326, eq. (9), for dμ read μdμ. P. 357, eq. (B), for dμ read dp'. STATICS. CHAPTER XIII. NON-COPLANAR FORCES. H ARTICLE 198.] Resultant of any Number of Forces applied to a Material Particle. Let a force P, represented in magnitude and direction by 00′ (Fig. 228), act on a particle at 0; let Ox, Oy, and Oz, be any three rectangular axes drawn through 0; and let the angles, O'Ox, O'Oy, and O'Oz, which the direction of P makes with the axes of reference be denoted by a, ß, and y, respectively. From let fall perpendiculars, O'F, O'H, O'D, on the planes, yz, zx, and xy, respectively, and let the parallelopiped be completed as in the figure. Then the force 00′ may be replaced by the forces OD and OC, by the parallelogram of forces; and OD can again be replaced by OA and OB. Hence the force P is equivalent to the three forces P cos a along Ox, Fig. 228. The converse proposition is also evidently true-namely, that any three forces, OA, OB, OC, along Ox, Oy, Oz (whether these are mutually rectangular directions or not), give a resultant represented in magnitude and direction by the diagonal, 00′, of the parallelopiped determined by the forces. If several forces, P1, P2, ... P, act at O and make angles (a1, B1, 71), (a2, B2, Y2), ... (an, B., Yn) with the axes, let each of them be replaced by its three components along Ox, Oy Oz; and if ΣΧ, ΣΥ, ΣΖ denote the sums of the components along the axes, we shall have ΣX = P1 cos a1 + P2 cos a1⁄2 + + P12 cos an 2 n (1) and the whole system of forces will be replaced by the three forces, EX, Y, and EZ along the axes of x, y, and z. But the resultant of three forces in these directions is the diagonal of the parallelopiped determined by them. Hence, R being the magnitude of this resultant, R = √(EX)2 + (ΣY)2 + (ΣZ)2, and if 0, 4, 4, be the direction-angles of R, (2) 199.] Graphic Representations of the Resultant. Since the resultant of any two forces, OA and OB, acting at O is obtained by drawing from A a line, Ab, parallel and equal to OB, and joining O to b, it follows that if a particle is acted on by n forces, 041, OA2, OA3, ... OA,, the resultant is obtained in magnitude and direction by drawing 4, 42 parallel and equal to OA2, a2 a parallel and equal to OA3, ... a-1 a, parallel and equal to OA, and joining O to a,; or, in other words, the side Oa which closes the polygon 04,a,a,... a, represents the resultant in magnitude and direction. When the sides of the polygon are not all coplanar, the figure is called a gauche polygon. Thus the second graphic representation of the resultant of a system of coplanar forces, which has been given in p. 19, vol. i, is equally applicable to non-coplanar forces. Hence, of course, it follows that a particle acted on by any set of forces which are parallel and proportional to the sides of a gauche polygon taken in order is at rest. Again, since by the parallelogram of forces, the resultant of OA, and 04, is 2. Og1, where g1 is the middle point of A12; and since the resultant of 2 Og, and OA, is 3 Og2, where 92 is determined exactly as in Art. 23, it follows that Leibnitz's graphic representation of the resultant is applicable to non-coplanar forces. This result follows also analytically; for if (X1, Y1, Z1), (X2, Y2, Z2), ••• (Xn, Yn, Zn) be the co-ordinates of the extremities |