Royal Society. 59 



Royal Institution, proposed to explain the extreme darkness of the 

 dots, under certain conditions of focus and illumination, by the hy- 

 pothesis that some of the oblique rays are thrown out of the field by 

 internal reflexion, being incident at the upper surface at an angle 

 too large for emergence ; but this does not appear to invalidate the 

 present hypothesis respecting the course of the transmitted rays. 



It does not appear to be desirable that objects should be illumi- 

 nated by an entire, or, as it may be termed, a solid cone of light of 

 much larger angle than that of the object-glass. The extinction of an 

 object by excess of illumination may be well illustrated by viewing 

 with a one-inch object-glass the Isthmia illuminated by Gillett's 

 condenser. When this is in focus, and its full aperture open, the 

 markings above described are wholly invisible ; but as the aperture 

 is successively diminished by the revolving diaphragm, the object be- 

 comes more and more distinct, and is perfectly defined when the 

 aperture of the illuminating pencil is reduced to about 20°. The 

 same point may be attained, although with much sacrifice of defini- 

 tion, by gradually depressing the condenser, so that the rays may 

 diverge before they reach the object ; and it may be remarked gene- 

 rally that the definition of objects is always most perfect, when an 

 illuminating pencil of suitable form is accurately adjusted to focus, 

 that is, so that the source of light and the plane of vision may be 

 conjugate foci of the illuminator. If an object-glass of 120° aper- 

 ture or upwards be used as an illuminator, the markings of Diato- 

 macese will be scarcely distinguishable, with any object-glass ; the 

 glare of the central rays overpowering the effects of structure on 

 those that are more oblique. 



" On the Formation of Powers from Arithmetical Progressions." 

 By C.Wheatstone, Esq., F.R.S. 



The same sum n a may be formed by the addition of an arithmetical 

 progression of n terms in various ways. Hence we are enabled to 

 construct a great variety of triangular arrangements of arithmetical 

 progressions, the sums of which are the natural series of square, 

 cube and other powers of numbers. Among these there are several 

 which render evident some remarkable relations. 



Each of the following triangles is formed of a series of arithmeti- 

 cal progressions, the number of terms increasing successively by 

 unity. 



The first term of an arithmetical progression of n terms having a 

 common difference S, and whose sum is n", is equal to 



»(«-i)+|(l-n). 



§ 1. SQUARE NUMBERS. 

 If S = n 5 , the first term =n+Ul— »). 



A. 

 Every square n 2 is the sum of an arithmetical progression of n 

 terms, the first term of which is unity and the difference 2. 



