2 REPORT — 1844. 



Again, let there be three events ; then replacing p^, p„, p-^, by the combinations 

 Pi P-2' Pi jPs' Pi Ps' ^^ have, by the same principle, 



PiPi'^'PiPa'^'PiPs^^ prob. of one of these compound events at least. 

 + prob. of two at least. 

 4- prob. of all three. 

 But the combination of two of the compound events, at least, is obviously the same 

 as the combination of all the simple events ; so is the combination of all the com- 

 pound events, and the probability of all the simple events happening is known to be 

 Pi Ps P3 ' hence 



'PiP2+PiP3+P2P3= prob. of two of the events at least happening together, 



+ '¥lP".P3> 



•*• PiP2'^PiP3'^P2Pa~^PiP2P3=^P^o^- of *^wo of the events, at least, happenmg 

 together. 



Moreover, the probability of one of the events, at least, happening is, by the prin- 

 ciple, equal to the sum of the individual probabilities diminished by the expression 

 just deduced, and by PiP^Pz ; that is, 



Pi'^P2'^Pi~PiP2~P\'P3~P2pi'\-PiP2P3= prob. of one, at least, of the three events 

 happening. And so on. 



On the Summation of Infinite Series. By Mr. Rawson. 



Tliis was a mode of combining the theorems of Laplace and Taylor in such a 

 manner as to render series very rapidly convergent, so as greatly to facilitate the cal- 

 culation of tables, and to render other arithmetical processes more convenient than 

 at present. Mr. Hodgkinson, who communicated the paper, pointed out its impor- 

 tant relations to some of the more general processes of integration. 



On the Double Square Representation of Prime and Composite Numbers. 



By J. J. iSYLVESTER, M.A. 



He first alluded to what had been done by the French mathematicians ; and then 

 pointed out the manner in which he thought numbers might be conceived to be com- 

 posed of squares ; and concluded by "mentioning some of the advantages which might 

 be expected from this mode of considering them. 



On a Theory of Quaternions. By Sir William R. Hamilton, 3I.R.I.A. 

 It has been shown, by Mr. Warren and others, that the results obtained by the 

 ordinai-y processes of algebra, involving the imaginary symbol V — l, admit of real 

 interpretations, such as those which relate to compositions of linear motions and ro- 

 tations in one plane. Sir W. Hamilton has adopted a system of three such imagi- 

 nary symbols, i,j, k, and assumes or defines that they satisfy the nine equations 



f =j' :=]r = — 1, ij =zk := — j i, jk-^i-=. — kj, k i ^j = — ik, 

 which however are not purely arbitrary, and for the adoption of which the paper 

 assigns reasons. He then combines these symbols in a quaternion, or imaginary 

 quadrinomial, of the form 



Q = ji' -^ i X -T- j y + kz, 

 in which w, x, y, z are four real quantities ; and states that he has established rules 

 for algebraical operations on such expressions, and has assigned geometrical inter- 

 pretations corresponding ; so as to form a sort of Calculus of Quaternions, which 

 serves as an instrument to prove old theorems, and to discover new ones, in the geo- 

 metry' of three dimensions, and especially respecting the composition of motions of 

 translation and rotation in space. 



An Account of the State of the Reductions of the Planetary and Lunar Ob- 

 servations made at Greenwich. By the Astronomer Royal. 



He announced that the planetary observations from 1750 to 1830 had been reduced 

 by the aid of Bessel's tables, and their places deduced and compared with those 

 given by the best tables for each planet ; and this portion vv^as complete. The print- 

 ing also was nearly finished. The reduction and comparison of the lunar observa- 



