1862.] 179 



1 1 . U and H, the body's single momentum and momentum couple, 

 may be found without difficulty in the ordinary way. In the case of 

 a rigid body moving about a fixed point O, if A, B, C be the moments 

 of inertia, and w x , ay w g be the angular velocities of rotation about 

 the principal axes O^,, O , O^, then H is the resultant of Aw^,, Boy 

 CM Z . Let then A denote the instantaneous axis and angular velocity 

 of rotation, then D t (H) is, by one of our fundamental theorems, 

 equivalent to det (A, H), together with 



I (A Wjt ) H to O a jf (B*,) || to <> | (Co,,) || to O,, 



and by D'Alembert's principle D t (H) is the axis of the resultant 

 couple formed by transferring to O all the external forces. 



12. The last theorem includes Euler's equations, and the different 

 extensions which those have of late received. I have attempted to 

 show that, in solving mechanical problems, the above theorem will be 

 generally found more useful than any of those equations. Moreover, 

 it serves to explain the geometrical reason, independently of Euler's 

 equations, why the results are so much simplified in the case of two 

 of the principal moments of inertia being equal to one another. 



13. In order to illustrate how advantageously and how completely 

 the most complicated formulae of dynamics may be interpreted, I 

 have given a direct analytical proof of Euler's equations, and have 

 then shown that the consideration of each step of the analytical 

 work leads to an extremely short demonstration of the same equations 

 by means of the theory of the differential coefficients and determinants 

 of lines. 



XV. " On the Theory of Probabilities." By GEORGE BOOLE, 

 Esq., F.R.S. Received May 21, 1862. 



(Abstract.) 



This paper has for its object the investigation of the general analy- 

 tical conditions of a method for the solution of questions in the 

 Theory of Probabilities, which was published in a work entitled "An 

 Investigation of the Laws of Thought, &c." * 



The application of the method to particular problems has been 

 * London, Walton and Maberly, 1854. 



