1086 HON. LORD M'LAREN ON SYSTEMS OF 



whence 



a 2 cos 4 + (a 2 + 6 2 )cos 2 0.sin 2 + 6 2 sin 4 = a?b 2 /r\ [a = £ ; 6 = £] . Dividing by a 2 6 2 , 

 .*. 9 cos 4 0+13 cos 2 sin 2 + 4 sin 4 = 1/r 2 . 



= 



0° 



5° 



10° 



15° 



20° 



25° 



30° 



35° 



40° 





r = 



•333 



•33 



4336 



•340 



•345 



•351 



•359 



•369 



•380 





= 



45° 



50° 



55° 



60° 



65° 



70° 



75° 



80° 



85° 



90° 



r = 



•392 



•406 



•421 



•437 



•452 



•467 



•480 



•491 



•498 



•500 



Eeturning to the equation of the wave-surface in its usual form (1), the curve of any 



section through an axis, Z, is most easily computed by transforming to polar coordinates. 



If, as usual, we make x = rcos0.cos(p ; 2/ = rsin0.cos<£ ; z=rsm<p; and then divide by 



v 2 cos 2 <£, we obtain 



/c 2 - r 2 \ ( /a 2 . cos 2 0\ , /6 2 sin 2 0\ ) . 



(^){(l*^) + \^¥)\= tan *' 



whence 4> may be found for any given values of r and . 



In the following illustration, PI. III. fig. (6), I suppose a section through Z making 



the angle 



= 45; sin0 = cos0= VF- C = 3; 6 = 2; a = l. 



The form of the equation shows that r must be >2 ; <3 .* 



T — 



20 



205 



21 



2-2 



23 



2-4 



25 



26 



2-7 



2-8 



29 



30 



6 = 



90° 



73°2 



58° 



47°-l 



39°-7 



33°8 



28°-7 



24°-2 



19°9 



15°5 



10°5 



o°- 



The two curves are shown in figs. (7) and (6), and although the first is of the 4th 

 degree and the 2nd is of the 6 th degree, the resemblance is very apparent. These may 

 be compared with the curve of PI. II. fig. (5), which represents the symmetrical equation 



cc 4 +a;y+2/ 4 = K2/ 2 +2/ 2 )- 



19. Curves Symmetrical about One Axis. 



It has been observed that an ordinary section of a central surface is only central when 

 it is taken parallel to a principal plane. But now, if a central section be taken in any 

 direction through an axis of symmetry (Y) of the central surface, then all sections parallel 

 to this will be symmetrical about y, but will not necessarily or usually be symmetrical 



9 — r 2 1 8 — 2r 2 

 * The numerical equation is -— == — — + — -= — — = tan 2 ? . 

 * 18^-18 9r 2 - 36 



