84 KANSAS Academy of Science. 



lotions of salicylic, boracic, and similar acids, naphthalene and the metallic chlo- 

 rides, separately and in combination, and in various strengths of alcohol and 

 glycerine. 



With these fluids, tests are being made by immersing insect larvae and imagines, 

 of all of which specimens careful notes respecting color and measurements are made 

 before immersion in the fluids. Inspection is made frequently, and any changes in 

 the specimen are noted. Especially valuable results are expected from simultaneous 

 experiments on a large number of larvje of the same species of lepidoptera. For 

 example, a number of larvse of an Arctian moth, all taken at one time and all of 

 nearly equal age, introduced into a series of different fluids, show especially well by 

 comparison the effects of the various preservative agents. 



The experimenting was begun too recently to allow of any collaboration of re- 

 sults thus early. At a future meeting of the Academy, the experimenters hope to 

 present the result of their work. 



IMAGINARY FOCAL PROPERTIES OF CONICS. 



BY H. B. NEWSON, LAWBENOE. 



In modern geometry, the foci of a curve are defined as the points of intersection 

 of the tangents to the curve from the imaginary circular points at infinity. Since 

 four such tangents can be drawn to a conic, and since four lines intersect in six 

 points, it would seem that everj' conic has six foci. But the circular points at infin- 

 ity count for two ; so there are only four finite foci, two real, and two imaginary. 

 Many authors refer to the existence of the imaginary foci. C. Smith, in his conic 

 sections, page 205, has shown that the imaginary foci are a pair of conjugate im- 

 aginary points, situated on the minor axis equi-distant from the center. Else- 

 where (Annals of Math., vol. V., page 1)1 have called attention to the fact that 

 imaginary focal properties of conies may be developed which are the counterparts 

 of the real focal properties. 



I propose in this paper to set down in parallel columns the real and imaginary 

 focal properties of conies, so that the reader can see at a glance what theorems 

 concerning imaginary foci correspond to known theorems concerning real foci. 

 I shall give first the theorems for the ellipse. They are given without proof. The 

 simplest method of proof for the imaginary properties is to change a into b, and 

 X into y, in the ordinary demonstration of the real properties. 



BEAL. 



The sum of the distances of any point 

 on an ellipse from the real foci is con- 

 stant, and equal to 2a. 



la' — f' 



The real eccentricity is e= ,/ . 



•■^ a^ 



The distance of either real focus from 



IMAGINABT. 



The sum of the distances of any point 

 on an ellipse from the imaginary foci is 

 constant, and equal to 26. 



The imaginary eccen- ^52 ^^2 



tricity is e=,/ . 



The distance of either imaginary fo- 



the center is ae. cus from the center is be. 



The length of the real semi-latus rec- The length of the imaginary semi- 



tum is a(l — e-). 



The lengths of the real focal radii 



latus rectum is 6(1 — c^). 



The lengths of the imaginary focal 



are a±ex. radii are b + ei/. 



