SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1071 



and for the sum of the 2nd and 3rd terms, 



1/4P {x 6 - x*y 2 - x*y* + y 6 } = PX 4 Y 2 + PX 2 Y 4 ; 



whence, by addition and subtraction we find for (a') and (ft) (being the curves, a and ft, 

 referred to their secondary axes). 



(l + P)x 6 +(15-P)*Y+(15-P)xY + (l + P)2/ 6 = 4A 6 . . . (a\); 

 (l-P> 6 +(15+P>Y+(15 + P)xY + (l-P)2/ 6 = 4A 6 . . . (ft). 



Again, in the transformation of (e) and (£) we have for the sum of the two extreme 



terms, 



l/4{6x 5 2/ + 20xhf + 6xy 5 } = X 6 - Y 6 , 



and for the sum of the two intermediate terms, 



1/4P{ =F 2x 5 y ± 4&y =F 2xy 5 } =F PX 4 Y 2 ± PX 2 Y 4 , 



where the upper and the lower signs belong respectively to the equations (e) and (£). 



Observing that the numerical coefficients are divisible by 2, we have for (e) and (<^), 



by addition, 



(3-~P)x 5 y + (l0 + 2V)x 3 y 3 + (3-?)xy 5 = 2A 6 (e) 



(3 + P)x 5 2/+(10-2P)a;Y + (3 + P)^ 5 = 2A 6 .... (O 



The above are all the forms of symmetrical diametral equations that can be formed with 

 four or three terms. If we seek for those that may be formed with only two terms, it is 

 evident that 



(1) The form x 6 + y 6 = A 6 , is a limiting form of (a) and (ft when P = . (c^) ; 



(2) x 6 — y 6 = A 6 , is a limiting form of (e) and (£) when P = . . . (e 1 ) ; 



(3) 4x Y + 4#Y = A 6 , is a limiting form of (ft), and therefore of (ft when P = l, 



and is also a limiting form of (a'), (a), when P = — 1 . . . (ft) ; 



(4) ix b y + 4xy 5 = A 6 , is a limiting form of (<^), (£), when P = 5, and of (e), (e) 



when P=-5 (f t ). 



Now we cannot directly obtain the last two forms with the negative sign from any of 

 preceding equations. Hence, there are apparently two independent limiting forms, 



AxY— 4xY = A 6 ; (e") and ixhj - Ixtf = A 6 . '. . (ft'). 



On further consideration, it is seen that (e") is derivable from (e) or (e), if the 

 coefficients of the intermediate terms in the fundamental equations are supposed invari- 

 able, while the extreme terms, x 6 , y 6 are supposed to be multiplied by coefficients which 

 are indefinitely diminished. Transforming to secondary axes, we find 



(1) The equation of (e") is unaltered in form and value by transformation (e") 



(2) The equation of ft' is 2x h y-±xhf + 2x'y h =\ (ft'), 



