k 



OF SYMBOLICAL GEOMETRY AND MECHANICS. 499 



The length of this small line is rd9, assuming d9 = angle PAF ; but the direction unit of a 

 line is expressed by dividing the symbol of the line by the symbol of its length ; hence the direction 

 unit of the small line is 



rde de 



'rid °' W 



de 



Hence the direction unit of a perpendicular to a line re is — . 



du 



In the Paper already referred to, which was read before the Society some months since, the 



reader will find this method of representing perpendicularity by the differential notation fully 



developed, and the notation Du.u, thence derived, explained, together with an auxiliary notation, 



Au . ii' ; both of which we shall have occasion to make use of hereafter. 



11. The following is the method we shall adopt of representing curves and surfaces sym- 

 bolically. 



To represent a curve or line we shall suppose a variable parameter to be involved in the symbol 

 of a point, in which qase it is clear, that the point will be indeterminate in position, but restricted 

 so far, that it will always be found upon some curve or line. The symbol of a point therefore, 

 when it involves a variable parameter and is thereby made indeterminate, becomes a symbolical 

 formula defining some line or curve, and may be called the formula of that line or curve. 



In like manner the symbol of a point, when it involves two variable parameters, becomes a 

 symbolical formula defining some surface, and may be called the form?ila of that surface. This 

 virtually amounts to defining lines and surfaces by symbolical polar equations. 



It is important, however, to observe that we suppose the variable parameters here spoken of to 

 be number symbols. If the variable parameter be a direction unit, it must be regarded as equi- 

 valent to two number symbols. 



12. The following are examples of this method of representing curves and surfaces. 

 The general symbolical formula of a straight line in space is 



u + re, 

 where u is the symbol of a given point, r a numerical variable parameter, and e a given direction 

 unit. 



For take OA = u (O being the origin) OB = e, draw a line through A 

 parallel to OB, taking upon it AP equal in length to r. Then AP is repre- 

 sented by the symbol re, and therefore m -t- re is the symbol of the point P, 

 which, since r is indeterminate, niay be any point of the line drawn through A 

 parallel to OB. 



It appears, therefore, that u + re is the formula a straight line drawn through the point whose 

 symbol is u, in the direction represented by e. 



1.3. In like manner the general symbolical formula of a plane is 



u + re + r'e' 

 r and r being numerical variable parameters. 



For take OA = u, OB = c, OC = e, draw AP parallel to OB and equal in 

 length to r, PQ. parallel to OC and equal in length to r'. Then, it is evident, 

 that M + re + r'e' is the symbol of the point Q; and that, since r and r are 

 indeterminate, Q is any point of the plane which contains the point A and is 

 parallel to OB and OC. 



Hence n + re + r'e' is the formula of a plane which is parallel to the directions represented by 

 t and e', and contains the point whose symbol is u. 



