326 BROOKS — ORTHIC CURVES. [May 20, 



We can'choose the ^'s in such a way that the pencil will include 

 the given curves, (i) and (2), for the 4n — 2 equations 



a\. = a,, -f ^2^\, v = i » . . 2n — I 



just suffice. We must show now that when the coefficients are deter- 

 mined as above, all the curves of the pencil are real. 

 Now we have 



and 



«2n-v = ^2n-v + A^'211-v 



From these, we get '. 



I a^ + A«'v = «v = «2n-vA \ = ^2n-vA ^ + ^'2n-y, 



and therefore 



a^ — ^'sn-v = (^2n-v ^'v)A~^- 



But this is the condition that every curve of the pencil be real. It 

 is clear that no curve not orthic can be included in the pencil. So 

 we see that : 



Any two real orthic curves of order n determine a pencil of real 

 curves of the same order ^ all of which are orthic. 



VIII. The Serret Circle, or Locus of Centres. 



M. Serret's theorem (Part Three, IV) on the locus of centres is 

 easily verified. The centre of any curve of the pencil is 



x = ^{a,^ta\^. 



Now if / is regarded as a parameter, this is the map equation of a 

 circle with its centre at 



The locus of centres of the 77iost general pencil of orthic curves is a 

 circle. 



In the special case where n of the intersections of the pencil are 

 at infinity, the locus of centres degenerates into a right line. A 

 pencil of this type may be written 



