and Loci of Apollonius, ^c. 43 



JTG is known, and the other point H in which UA cuts 

 circle JTGU, and .•. the point C in which the circle II AO 

 cuts MM, and hence CAI and BID, 



The composition, &c., may be easily made. However, in 

 order to familiarise the methods of arriving at the limits of 

 angular magnitudes in other questions, I Avill indicate the 

 natui'e of the limits of 6 right. 



Limiting Values for 6 Right. 



As the points O and A are known independently of ^, .•. it 

 is evident tliat by describing the two circles through O and A 

 which touch MINI, and putting H' and H' for the points in 

 which they again cut circle JTG, and U' and U' for those in 

 which Air and AH' again cut this same circle, and W' and 

 W for the points in which GU' and GU' cut NN, then will 

 the angles AV'G right to N and WG right to N be the limiting 

 values of 6 right. And, if h' and to' be the points in Avhich 

 GA cuts the circle JTG and line NN, it is evident that G 

 may be regarded as a position of U corresponding to to'. 



Moreover, it is evident that if iv' lies between W and W, 

 and that h' is not inside or outside both circles through A and 

 O touching MM, then Avill the circle OAh' cut MM in imagi- 

 nary points; and the limiting values for right are evidently 

 such as to include between them all values of right (and no 

 others) for which the lines Al and BI are imaginar3^ But if 

 iv' lies between W and W, and that li is inside or outside 

 both circles through A and O touching MM, then will the 

 limiting values of d right be such as to have outside them all 

 values of right (and no others) for which the lines Al and 

 Bl are imaginary. 



And it is further evident, that if iv' lies outside WW and 

 that h is not inside or outside both circles through A and O 

 touching jMjSI, then Avill the limiting values of 9 right be such 

 as to have outside them all values of B right (and no others) 

 for Avhich Al and Bl are imaginary; but if w lies outside 

 WW, and that li is inside or outside both circles through A 

 and O toiiching !MM, then will the limiting values of B right 

 be such as to include between tliem all values of right (and 

 no others) for which Al and Bl are imaginar3^ 



When Q right is equal either limit, the lines Al arc coinci- 

 dent and real. 



Moreover, it is evident that when A and O are not on the 

 same side of MM there arc no real Hmits to angle right, &c. 



