816 PROFESSOR CHRYSTAL ON THE 



whence 



2udu dg _ 

 n i + 2-Sv + J~ U; 



that is, 



The integral of (4) is 

 Now (5) may be written 



( * 2 Wf =0 . . . (4). 



\W— 2 W — 1/ £ 



(«-2)V^/(«-l)=- 4c < 5 )- 



or 



Hence 



(y _ 4. u + 4.) x + 4 ctt _ 4 C = o , 

 xu 2 — 4(« — c)u + 4(« — c) = 



X ' 



whence, squaring and substituting the value of a, we get 



Sy=-(x-c)(x-2c)±2c\c-xf (6); 



or, in rational form, 



(x 2 + 12y)c 2 -2x(x 2 + 9y)c + (x 2 + Syf = .... (7). 



The c-cliscriminant is therefore 



x\x 2 + Q y y - ( x 2 + 1 2y)(x 2 + 2,yf = , 

 which reduces to 



*/ 3 = (8). 



The primitive curve can be readily traced from (6). 



We observe, in the first place, that the value of y is unchanged if we change the 

 signs of both c and x. It follows that the curve of the family for any negative value of 

 c is the image in the ?/-axis of the curve for the corresponding positive value of c. We 

 may therefore confine our attention to positive values of c ; and we also see that, for 

 real values of y, x must not exceed + c. 



Since 2c - x <£ 2 J{c(c -x)}, the value of y is negative for all admissible values of x t 

 and vanishes only when x — and x = c. 



From (6) we have 



7/ =^Zc-2x)?r s l{c(c-x)} .... (9); 



f=-%±hJ{cl(c-x)} ... . (10). 



If we speak of the parts of the curve corresponding to the upper and lower signs as the 

 first and second branches, we see at once that for the second branch y' is always positive 

 and y" always negative. This branch crosses the i/-axis at (0, - % c 2 ) and is uniformly 

 convex to the .x-axis. 



