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145] A MEMOIR UPON CAUSTICS. 377 
equations which give 
Z 2 Yx = Y (mZ 2 — AX 2 ), 
- Z 2 Yy = Z 3 + X ('mZ 2 - AX 2 ), 
Z 4 Y 2 (x 2 + y 2 - a) = 4AZ 3 XY 2 , 
whence eliminating, we have 
{Z 3 + X (mZ 2 - AX 2 )} 2 + Y 2 (mZ 2 - AX 2 ) 2 - Z 3 Y 2 (aZ + 4AX) = 0, 
where if, as before, c denotes the radius of the refracting circle, a the distance of the 
radiant point from the centre, and ti the index of refraction, we have 
a = - a 2 + (1 + -\) c 2 , 
/a 2 y? 
A = 
c 2 a 
2/J 2 ’ 
The above equation is the condition in order that the line Xx + Yy + Z = 0 may be 
a normal to the secondary caustic (x 2 + y 2 — a) 2 + 16A (x — ??i) = 0, or it is the tangential 
equation of the caustic, which is therefore a curve of the class 6 only. The equation 
may be written in the more convenient form 
Z 6 + 2Z S X (mZ 2 - AX 2 ) + (X 2 + Y 2 ) (mZ 2 - ^1X 2 ) 2 - Z 3 Y 2 (ocZ+4AX) = 0. 
XXXVIII. 
To compare the last result with that previously obtained for the caustic by 
reflexion, I write ^ = — 1, and putting also c = l and Z=a (for the equation of the 
reflected ray was assumed to be Xx + Yy + a — 0), we have 
a = a 2 + 2, A — \a, m = ^ (1 + 2a, 2 ), 
and the equation becomes, after a slight reduction, 
4a 4 + 4a 2 X (2a 2 + 1 - X 2 ) 4- (X 2 + F 2 ) (2a 2 + 1 - X 2 ) 2 - 4a 2 F 2 (a 2 + 2 + 2X) = 0, 
which may be written 
(2a 2 + X (2a 2 + 1 - X 2 )) 2 + F 2 (- 4a 2 + 1 - 8a 2 X - 2 (2a 2 + 1) X 2 + X 4 ) = 0 ; 
this divides out by the factor (X + l) 3 , and the equation then becomes, 
(X 2 - X - 2a 2 ) 2 + F 2 ((X - l) 2 - 4a 2 ) = 0, 
which agrees with the result before obtained. 
C. II. 
48