PHILOSOPHICAL SOCIETY OF WASHINGTON. et 



be the same as the number of times that the focal curve intersects 

 the focal plane. But the focal plane is necessarily parallel to 

 the axis of abscissas ; and, therefore, the greatest possible num- 

 ber of intersections of the curve with the plane can only exceed 

 by one the number of tangents which can be drawn parallel to 

 the axis of abscissas. To find these tangents we equate (6) to 

 zero, and obtain 



= mD>r-' + wEjt"-i -f etc. (t) 



As % can never be either zero, negative, or imaginary, we have 

 to consider only the real positive roots of this equation'; each of 

 which corresponds to a tangent. To make the number of tan- 

 gents as great as possible the quantities D, B, F, etc., must be 

 independent of each other ; which will be the case when'the right 

 hand members of the equations (4) contain as many As as there 

 are powers of % in the dispersion formula (4). All the terms of 

 (7) contain the common factor x°'-\ Taking it out we have 



— mJ) = uEtJ'-'^ + pFfT-"' + etc., (8) 



from which it is evident that the number of real positive roots in 

 (7) will always be one less than the number of powers of si in (3). 

 Hence we conclude that — 



In any system of infinitely thin lenses in contact, the number 

 of lenses required to bring the greatest possible number of light- 

 rays of different degrees of refrangibility to a common focus is 

 the same as the number of difl"erent powers of % contained in the 

 dispersion formula employed. 



The method made use of in arriving at this result has been 

 adopted, because it brings out clearly the geometrical relations 

 of the problem. The result itself is evident from a mere inspec- 

 tion of equation (5), which cannot possess more real positive 

 roots than it has independent auxiliaries D, E, F, etc. 



The communication of Mr. Button 



ON THE PERMIAN FORMATION OF NORTH AMERICA 



then followed. 



Mr. Button stated that many geologists have long been in 

 doubt whether the Permian formation was merely of local occur- 

 rence in a very few districts constituting a subordinate series 

 embraced within and forming a part of the closing period of the 

 Carboniferous series, or whether it was of world-wide prevalence 



