SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 



1077 

 The 



there is a series of equivalent curves which go through corresponding phases, 

 different forms are shown in PL II. figs. 1 to 4. 



In their complete forms the equiaxial curves (e) of the 6th degree have three pairs of 

 asymptotic axes ; one pair being the secondary axes, whose inclination to the primary 

 axes is always 45°; the other two pairs having an inclination depending on the value 

 of P (fig. 3). 



Table of Computed Places. 





Curve 

 11. 



e 



X 



y 



10° 



1005 

 0-177 



15° 



1012 

 0-271 



20° 



1021 

 0372 



25° 



1034 

 0-482 



30° 



1054 

 0-608 



35° 



1079 

 0756 



40° 



1146 

 0961 



44° 59' 



2470 

 2-468 



45° 



OO 

 00 



Curve 

 12. 



e 



X 



y 



2° 



1002 

 0035 



4° 



1013 

 0071 



6° 



1030 

 0108 



8° 



1059 

 0149 



10° 



1106 

 0195 



12° 



1190 

 0253 



14° 



1-414 

 0353 



15° 



00 

 00 



Curve 

 13. 



e 



X 



y 



10° 



1000 

 0176 



15° 



1000 

 0-268 



20° 



1001 

 0364 



25° 



1002 



0-467 



30° 



1007 

 0-581 



35° 



1021 

 0715 



40° 



1074 

 0901 



44° 59' 



2-461 

 2460 



45° 



OO 

 OO 



Curve 

 14. 



6 



X 



y 



5° 



0-9986 

 0-0874 



10° 



0-9951 

 0-1755 



15° 



0-9895 

 02652 



20° 



0-9824 

 03576 



25° 



09754 

 0-4548 



30° 



09721 

 05611 



35° 



0-9797 

 06860 



40° 



1024 

 0861 



45° 



OO 

 OO 



Curve 

 15.* 



e 



X 



y 



log r 



0° 



00 



0° 



00 



2° 30' 



1-338 

 0-058 

 0127 



5° 



1191 

 0104 

 0-078 



7° 30' 



1113 

 0-147 

 0050 



10° 



1060 

 0187 

 0032 



12° 30' 



1021 

 0226 

 0019 



15° 



0989 

 0265 

 0010 



17° 30' 



0-964 

 0304 

 0005 



22° 30' 



0924 

 0383 

 00011 



Curve 

 16. 







X 



y 



0° 



00 







5° 



1-786 

 0156 



10° 



1-421 

 0251 



15° 



1-247 

 0334 



20° 



1139 

 0414 



25° 



1066 

 0-497 



30° 



1020 

 0589 



35° 



1000 

 0700 



40° 



1030 



0-865 



45° 



OO 

 OO 



In the form (e), when P = 3, the curve has the limiting form of the equilateral 

 hyperbola (of 2nd degree), the three pairs of asymptotes being there coincident. 



When P>3, there are six equal real branches, and six conjugate branches. If P 

 exceeds 3 by a very small quantity, the first real branch (bisected by X) is nearly rectan- 

 gular, and the first conjugate branch is extremely acute. The secondary axis divides 

 this from a similar acute real branch ; and then there is a nearly rectangular conjugate 

 branch bisected by Y. When P = 7, the inclination of the first pair of asymptotes is 



* These values of log r are identical with those of No. 10 of the preceding table. In No. 15 the curve is referred 

 to asymptotes. 



