84 Improvements in Fundamental Ideas 



it is evident that, according as the sign of - is positive or 

 negative, so must the directions PA and QB be like or un- 

 like when regarded as relative to directions of same species 

 on MM and NN respectively ; it is also plain that if — be 

 restricted to one sign, there is but one answerable point A, 

 one circle AIB, two points E, and two solutions to the 

 problem. 



But if - be unrestricted as to sign, there are two answer- 

 able points A, two circles AIB, and therefore- four answer- 

 able points E, and four solutions. Moreover, since the two 

 points A must be on opposite sides of MM, one of the circles 

 AIB will always cut MM in real points E, the other two 

 points E being real or imaginary, just according as the other 

 circle AIB cuts MM in real or imaginary points. 



We will now find the limiting values for the angular mag- 

 nitude right, and for the ratio ™, so as to be enabled so say 

 ' a priori' when the points E are real, imaginary, &c. 



Limiting values for ' 6 right.' 



To find the limiting values for the angles ' 6 right/ when 

 the rest of the data is unchangeable, we may proceed as 

 follows : — 



The point A is evidently fixed independent of 6 right. 

 Looking on the triangle DEO, we see the angle DO right 

 to E is constant for all values of 9, and it is evident that 

 when OE right to D or OB right to C is at its limit, then 

 will ED right to O or EA right to B be at its limit ; but 

 this last angle is evidently at a limit when the circle BAE 

 touches MM. Hence it is evident that, by describing the 

 two circles through A and B, which touch MM, and putting 

 i and i for the other points in which they cut circle ACH, 

 then will the angles i B right to C, and i B right to C, be the 

 required limits. And it is moreover evident that, according 

 as the given magnitude of ■ 6 right ' is not comprehended 

 between these limits, or equal to one of them, or compre- 

 hended between them, so accordingly will the corresponding 

 circle AIB cut MM in two imaginary, in two real and 

 co-incident, or in two real and distinct points, E. It is also 

 evident that should A and B not lie on the same side of 

 MM, then will the points E be real for all values of ' 6 

 right/ 



