2G6 Dr. L. Trenchard More on the Coincidence of 



By elimination of « and arrangement of terms, the equation 

 of the envelope may be written in the following form : — 



(a)...c 4 {x 2 -b^(z 2 -a 2 )-a\x 2 -c 2 ){z 2 - b 2 ) [(fig. 2). 



-f 2acxz \ / {c 2 (z 2 -a 2 )-a 2 (z 2 -b 2 )\^c 2 {x 2 -b 2 )-a 2 (x 2 -c 2 )} = 



The equation is of the eighth degree and symmetrical about 

 the axes. By a permutation of the letters the cases where 

 the circle is wholly in the ellipse and where the ellipse is 

 wholly in the circle are at once obtained : 



V{a?-c 2 )ty 2 -a 2 )-a\x 2 -b 2 )(y 2 -c 2 ) [(fig. 3). 



+ 2abxy \/ \ b 2 (f - a 2 ) - a 2 {y 2 -c 2 )\\ b 2 (x 2 - c 2 ) - a 2 {x 2 - b 2 ) \ = 



cXz*-b*)(f-a*)-b 4 (f-c*)(z 2 -a*) [(fig. 4). 



+ 2bcyz ^{c q (z^-b^)-b 2 (z^-a 2 ) } { c *(y*-a*)-b 9 (y*-c 9 )} = 

 Fiar. 4. 



To obtain the locus for a negative uniaxal crystal, b i; 

 equated with c in equation («), giving 



(* 2 - V) { 1/z* - (1/a 2 -f 1/b* -f 2/ab sjx*jj? - V) } = 0. 



The locus thus breaks up into two parts, a curve of the sixth 

 degree and a pair of straight lines, at the extremities of the 

 diameter b and perpendicular to the axis X, shown in figs. 5 

 and 6. 



