8 REPORT — 1865. 



On the Calculation of the Potential of the Figure of the Earth. 

 By W. H. L. Russell. 

 The object of this paper was to simplify and render symmetrical certain portions 

 of Professor O'Brien's investigations on the figure of the earth. In that paper the 

 reduction of the expression for the potential to a convenient form is effected by the 

 introduction of a discontinuous quantity ; the author of the present paper has 

 found that the required form is obtained much more shortly by dividing the original 

 definite integral into two parts, and then expanding separately. 



On the application of D'Alembcrt's Principle to the Rotation of a Rigid 3/ass. 



By Dr. Steyelly. 

 The author explained that the present method of applying D'Alembert's prin- 

 ciple to the investigation of the spontaneous axis assumed by a free, rigid mass, 

 under the action of force, in all the works he was acquainted with, led to what he 

 showed to be a false conclusion, viz. that that axis must be a principal axis of the 

 rigid mass. He showed how the error arose from neglecting, in applying the prin- 

 ciple of D'Alembert, to take into account not only that part of the motion of each 

 elementary part of the body which related to the magnitude of its motion, but also 

 that part which relates to its direction, and from which its centrifugal endeavour 

 at each instant arises. But if the force impressed tend to pi'oduce rotatory motion 

 round an unstable spontaneous axis, how can the present mode of applying 

 D'Alembert's principle lead to a true conclusion, when it proceeds on the method 

 of bringing the body into such a state that the equations of equilibrium (that is, of 

 no after-change) shall give the direction-courses of the axis ? 



On a Special Glass of Questions on the Theory of Provabilities. 



By Professor Sylvester, F.R.S. 



After referring to the nature of geometrical or local probability in general, the 

 author of the paper drew attention to a particular class of questions partaking of 

 that character in which the condition whose probability is to be ascertained is one 

 of pure form. The chance of three points within a circle or sphere being apices of 

 an acute or obtuse-angled triangle, or of the quadrilateral formed by joining four 

 points, taken arbitrarily within any assigned boundary, constituting a reentrant or 

 convex quadrilateral, will serve as types of the class of questions in new. The 

 general problem is that of determining the chance that a system of points, each 

 with its own specific range, shall satisfy any prescribed condition of form. For 

 instance, we may suppose two pairs of points to be limited respectively to segments 

 of the same indefinite straight line: the chance of their anharmonic ratio being 

 under cr over any prescribed limit will belong to this category of questions, to 

 which, provisionally, the author proposed to attach the name of form-probability. 

 In questions of form-probability, in which all the ranges are either collinear seg- 

 ments or coplanar areas, or defined portions of space, rules may be given for trans- 

 forming the data, so as to make the required probability depend on one or more 

 probabilities of a simpler kind, leading to summations of an order inferior by two 

 degrees to those required by the methods in ordinary use. Thus Mr. Woolhouse's 

 question relating to the chance of a triangle within a circle or sphere being acute 

 can be made to depend upon an easy simple integration, the solutions heretofore 

 given of this problem involving complicated triple integrals. It was shown, as a 

 further illustration, that the form -probability of a group of points all ranging over 

 the same triangle remains unaltered when the range of one of them is limited to 

 any side of the triangle chosen at will, and, again, (for convenience of expression 

 distinguishing the contour into a base and two sides) will be the mean of the two 

 probabilities resulting from limiting one point to range over either side with uni- 

 form probability, and simultaneously therewith a second point of the group over 

 the base, with a probability varying as its distance from that end of the base in 

 which it is met by the side. An analogous rule can be given for transforming the 

 form-probability of a group limited to any the same parallelogram. So again for 

 a group of points ranging over a plane figure bounded by any curvilinear contour. 



