Mr. De Morgan on the relative Signs of Coordinates. 2b 1 



f and r) are both positive ; when in A' O B', f is + and >) — ; 

 when in B' O C, f and ij are both negative ; when in C O D', 

 f is — and >) + . If we now take the equations of transforma- 

 tion in the form in which they are most usually given, 



^ = ?j sin 9 + m cos 3 1 .on 



r; = « cos 9 — m sin fl j 

 we find these to be universally true on the conventions above 

 mentioned, and not on any others. Let PO A = a, and we 

 have 



f = -^^ cos (a — 9) »j = — — sin (« — 6) (3.) 



cos a cos a 



or (a < ^ tt) ^ is positive from 9 = fw + a to^^^ + ^j 

 and wzce versa ; while ij is positive from 9 = 7r + ato9 = a, 

 and vice versa. The other cases with respect to « easily 

 follow. 



To return to equation (1.), taking the line y = ax parallel 

 to j/ = a X -{■ b as yO a., the axis of ^, in which case 



, n — a in 



1 = ± / r , » 



V\+a^ 



this expression must by what precedes be interpreted as ne- 

 gative whenever O « falls between O A' and O C, and vice 

 versa. But as one part of the line j/ = ax falls somewhere 

 in the angles A' O B', B' O C, the preceding expression, ab- 

 solutely considered, has either sign. But if we attribute one 

 sign rather than the other to the corresponding value of ^, 

 we are then, simultaneously with this hypothesis, under the 

 necessity of assigning the proper sign to rj, as above deter- 

 mined. It must be observed that in the preceding hypothesis, 

 if f be measured positively in the angle AOA', ij is positive; 

 while if be measured negatively in that angle, >) is negative. 

 Similar considerations apply to the whole perpendicular p, 

 which is either positive or negative, accoi'ding to the hypo- 

 thesis made respecting the' positive and negative direction of 

 the line y = a x, and follows the same laws as any line which 

 is made up of the sum or difference of coordinates with re- 

 spect to given axes and assigned directions of positiveness or 

 negativeness. 



In tlie preceding considerations will be found the answer 

 to the question relative to the sign of the radius of curvature in 

 a curve. There is no such sign, except it be one of perfectly 

 independent convention, until it is settled which is the positive 

 and which the negative direction on the tangent. But the 

 preceding will be sufficient to illustrate my view of the sub- 

 ject as contained in the following assertion : that relatively to 



2F2 



