﻿34 Mr. G. F. H. Smith on the General Determination 

 Arrange (11) thus 



^4 _ ? 2{ 12 Q? + c 2) + m 2( e 2 + fl 2) + n 2( fl 2 + J2) } + 7^2,2 + ^2^2 + n l a «J> = Q 



diD 

 and differentiate twice, remembering that -~- vanishes for 





the critical values. 



m 



dm 1 

 d*n fdn 



p ^ i 2p*-l 2 (b 2 + c*) -m 2 (c 2 + a 2 )-n 2 (a 2 +& 2 )} -p 2 [(& 2 + c 2 ) [ I 



—{<£+©'} ~ w+ *{-£+(£) , !-° 



Putting p = b and m = 0, we have 



6~£ . {26 2 -Z 2 (6 2 + c 2 )-n 2 (a 2 + 6 2 )} = (a 2 -&2)(7> 2 - c 2 ). 

 dm* c 



Suppose & is greater than a and c. The term on the right is 

 negative, and the expression within the bracket on the left, 

 which can be arranged thus l 2 {b 2 — c 2 ) + n 2 (b 2 — a 2 ), is positive. 



Hence -r^ is negative, and, as indeed was otherwise obvious, 



the greatest value of p is a maximum value of the corre- 

 sponding curve. Similarly the least value is a minimum 

 value. 



If now b is the mean value, the term on the right is 

 positive. Again, since m =0, l\ + nv = 0, and, therefore, 



Substituting these values in' the expression within the 

 bracket we have {6 2 -(a 2 X 2 + 6V 2 + cV)}/(v 2 + X 2 ). But 

 a 2 \ 2 + b 2 /j? + cV is the square of the semi-diameter parallel to 

 the edge of the prism. Hence, if the diameter be greater 

 than the mean axis of the ellipsoid, the value corresponding 

 to the latter is a minimum value ; if less, a maximum value. 

 Now at any point lying on the circular sections, the semi- 

 diameter is equal to the mean semi-axis. If, therefore, the 

 direction parallel to the edge of the prism lies towards the 

 greatest axis of the ellipsoid, the corresponding value is a 

 maximum ; but, if towards the least, it is a minimum ; if it 

 lies on either of the circular sections, the two curves touch. 



The two intermediate values may be discriminated by 

 determining the angles of extinction with regard to the edge 



