and Loci of ApoUonius, ^c. 87 



If the sign of m.n is unrestricted, there are four answerable 

 points and tlieir corresponding lines LL, &c.^ &c. 



NOTES. 



By means of the porisms now given, and those in the 

 Th'ansactions for 1859, we are enabled to give simple solutions 

 to the following interesting problems, viz. : — 



To construct a quadrilateral having its four sides passing 

 through given points, and two of its opposite angles on given 

 straight lines, and such moreover that the other two angles 

 shall be of given magnitudes, or be situated on given straight 

 lines, or such that one of them shall be of a given magnitude 

 and the other be situated on a given straight line. 



They can be easily reduced to the third problem in Trans- 

 actions for 1859. 



And from the porismatie state of the data in these problems 

 numerous interesting and useful porisms may be derived — 

 one of which is as follows : — 



Given a point P a^id two straight lines MM, NN ; then if 

 B and C be any points in these lines such that the angle PB 

 7'ight to C shall be of a given magnitude, another jwint Q may 

 be found such that the angle QB right to C shall be of a con- 

 stant determinable magnitude. 



FINAL OBSERVATIONS. 



In the preliminary observations I have pointed out advan- 

 tages to be derived from my new methods of defining angular 

 magnitudes ; and those avIio have carefully studied the solu- 

 tions to the problems contained in this paper, cannot b6 

 unaware of the value of these advantages. However, I will 

 now give a few other instances of the improvements Avhieh 

 they eifect in important theorems : — 



First. The theorem on page 465 of the Geometric Supc- 

 rieure should be replaced by the folio Aving : — 



A moveable tangent tt rolling on the circumference of a 

 circle whose centre is C, cuts two fixed tangents m'm'. n'n', in 

 tivo points m, n, such that the "angle Cm right to u" is 

 constant = right. 



Second. The theorem on page 457 (which is so defective 

 that its reverse is imtrue) should be replaced by the foUoAV- 

 ins: : — 



