SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1051 



the plane xy ; <f> is the angular coordinate in the plane xz; y = xta,n0 ; z = xtsno.<f) ; and 

 the transformed equation is 



x 8 {l + tan 2 atan 2 + 3tan 2 atan^ + tan 8 0} = 1 



.-. - = { 1 -+- tan 2 0tan 2 + 3 tan 2 <9 tan*0 + tan 8 e} 1/8 . 



Suppose we want a series of values of x and y to the argument of tan<£=l or z = x, 

 the reduced equation is l/x= {2 + 4tan 2 #} 1/8 , from which the following values are directly 

 found : — 



e=°. 



15° 



30° 



45° 



60° 



Log tan 6 



= 1-4281 



1-7614 



00000 



0-2386 



Log tan 2 



= 2-8562 



1-5228 



00000 



0-4772 



Tan 2 



= 00718 



03333 



10000 



3-0000 



4tan 2 + 2 



= 2-2872 



33332 



60000 



140000 



Log(4tan 2 + 2) 



= 0-3593 



0-5227 



07782 



11461 



1/8 = log I/a 



= 00449 



0-0653 



00973 



01433 



Log a; 



= 1-9551 



1-9347 



1-9027 



1-8567 



Log y 



= 13832 



T6961 



1-9027 



00953 



X 



= 0-9018 



0-8604 



0-7993 



07190 



y 



= 02416 



07251 



0-7993 



5-2360 



(2) Let the equation be 



a 10 + /3V<5 3 + a 4 /3Y = l. . - (5) 



fi = ld; y = ma ; B = na; and the transformed equation is 



1 + £ 4 m 3 w 3 + l 3 w? = -r- • 

 a 



If it is desired to find values of a and 8 to the arguments 1=1, m=l, the reduced 

 equation is 2 + n 3 = l/a 10 [n = tan v.] 



V 



20° 



40° 



60° 



80° 



Log tan 3 t» 



= 1-5611 



1-9238 



0-2386 



07537 



Log tan 3 u 



= 2-6833 



1-7714 



0-7158 



2-2611 



2 + tan 3 u 



= 2-0482 



2-5907 



7-1980 



1844500 



Log (2+tan 3 y) 



= 03113 



04135 



08572 



2-2658 



1/10 = log l/o 



= 003113 



004135 



0-08572 



0-22658 



Log a 



= 1-96887 



T95865 



1-91428 



T-77342 



Log tan v 



= 1-5611 



1-9238 



0-2386 



07537 



Log 8 



= T5300 



T8825 



01529 



05271 



a 



= 09309 



09093 



0-8210 



05934 



8 



= 0-3388 



07630 



1-4220 



3-3660 



An equation consisting of a single homogeneous part may also be reduced to polar 

 coordinates and solved for r. If we write r cos 9 for x, and r sin 6 for y, and divide by 

 r n , the resulting equation is 



Cos re + A 1 .cos"- 1 0.sin0 + A 2 cos ra - 2 0.sin 2 0± . . . ±sin n = — = -, 

 VOL. XXXV. PART IV. (NO. 23). 7 X 



