134 Cambridge Philosophical Society. 



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



unbiassed, or if u= — , this becomes 

 t* 2' 



or « + ^^ —. 



2— fl 2— a 



For this writers on lo^c 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. 



Nov. 23. — On a New Notation for expressing various Conditions 

 and Equations 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, \du is the general expression for a right line 

 perpendicular to u. 



The sign \d 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 \d 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 Kdu denotes. This he does in the following manner. He 

 assumes 



u=^xa,-\-y^ + zy, 



(where a- ^ y reipresent three lines, each a unit in length, drawn at 

 right angles to each other, and if y z are any arbitrary numerical 

 coefficients,) and supposes that the differentiation denoted by d affects 

 a j3 y, but not x y z. This secures the condition that u is invariable 

 in length, and leads to the following expression for Kdu, viz, 



Xdu={zy' —z'y)(x, + (oaz' —x'z)^ + (yx' —y'a;)y, 

 x' y' z' being arbitrary coefficients. 



Assuming u'=x'a,+y'fi + z'y, it appears from this expression for 

 Xdu, that du=0 when u=u', and therefore that c? denotes a differen- 

 tial taken on the supposition that u' is constant. 



On this account the author substitutes the symbol Du' in place of 

 Ac? ; he then shows that the operation D^/ is distributive with respect 

 to u' (i. e. that Dm'+m"s=D„/4-Dm")> and to indicate this he elevates 



