33 



the falsehood of the conclusion. Various cases are examined ; but 

 it must here be sufficient to cite one or two results. 



If fj, be the probability which the mind attaches to a certain con- 

 clusion, a the probability that a certain argument is valid, and b the 

 probability that a certain argument for the contradiction is valid : 

 then the probability of the truth of the conclusion is 



. (1 -% 



(l-%+(l-«)(l-/*)- 



If b = 0, or if there be no argument against, and if the mind be 



unbiassed, or if u== — , this becomes 

 A 2 



1 . (l-«)°- 



or a+ v 



2 — a 2 — a 



For this writers on logic generally substitute a, confounding the 

 absolute truth of the conclusion with the validity of the argument, 

 and neglecting the possible case of the argument being invalid, and 

 yet the conclusion true. 



November 23, 1846. 



On a New Notation for expressing various Conditions and Equa- 

 tions in Geometry, Mechanics and Astronomy. By the Rev. M. 

 O'Brien. 



If A, P, P' be any three points in space, whether in the same 

 straight line or not, and if the lines AP and AP' be represented in 

 magnitude and direction by the symbols u and u', then, according to 

 principles now well-known and universally admitted, the line PP' is 

 represented in magnitude and direction by the symbol u' — u. Now 

 if AP and AP ; be equal in magnitude, and make an indefinitely small 

 angle with each other, PP' is an indefinitely small line at right angles 

 to AP, and u' — u becomes du. Hence it follows, that, if u be the 

 symbol of a line of invariable magnitude, du is the symbol of an in- 

 definitely small line at right angles to it ; and therefore, if X be any 

 arbitrary coefficient, Xdu is the general expression for a right line 

 perpendicular to u. 



The sign Xd therefore indicates perpendicularity, when put before 

 the symbol of a line of invariable length. The object of the author 

 is to develope this idea, and to show that it not only leads to a 

 simple method of expressing perpendicularity, but also furnishes a 

 notation of considerable use in expressing various conditions and 

 equations in geometry, mechanics, astronomy, and other sciences 

 involving the consideration of direction and magnitude. 



The author first reduces the sign Xd to a more convenient form, 

 which not only secures the condition that u is invariable in length, 

 but also defines the magnitude and direction of the perpendicular 

 which Xdu denotes. This he does in the following manner. He 

 assumes 



u=xa,+y(3-\-zy, 



