173 



May 12, 1856. 



A paper was read by Mr. Warburton " On Self-repeating Series," 

 in continuation of a former paper. 



The author showed in his former paper on self-repeating series, 

 printed in vol. ix. part 4. of the ' Transactions ' of the Society, that 

 in the fraction which generates a series of either of the following 

 forms, 



I 2n ±2 2n .t + 3 2n .t*±&c. . . , 

 or 



l 2re+1 + 2 2w+1 . t+Z 2n+l t*± &c. . . , 



the numerator of such fraction is a recurrent function of t. He also 

 then determined the coefficients of the several powers of t in such 

 numerator to be given linear functions of the differences (as the case 

 maybe) of 2n , or of 2n+1 . 



In his present paper, from the n pairs of equal coefficients which the 

 recurrent numerator contains, the author obtains n linear equations 

 between the 2n differences concerned ; and selecting any n of these 

 differences, he concludes that each of them can be expressed in terms 

 of the other n differences not so selected ; and consequently that no 

 formula, expressed in terms of the differences of 2n or 2n+1 , need 

 contain more than n of those differences. 



He gives the equations requisite for obtaining £. n+p (Q 2n ) in terms 

 of (A ra , A M-1 , . . . A*, A')0 2 ; and A w+1+ ^(0 2n+1 ), in terms of 



(A w+1 , A w , . . . A 3 , A2)0 2n+I ; and he applies these and other of his 

 equations to the elimination of particular differences of zero from 

 sundry formulas. 



Also, Mr. Bashforth exhibited models illustrating the Moon's 

 motion. 



Also, a paper was read by Mr. Maxwell " On the Elementary 

 Theory of Optical Instruments." 



The object of this communication was to show how the magnitude 

 and position of the image of any object seen through an optical in- 

 strument could be ascertained without knowing the construction of 

 the instrument, by means of data derived from two experiments on 

 the instrument. Optical questions are generally treated of with 

 respect to the pencils of rays which pass through the instrument. 

 A pencil is a collection of rays which have passed through one point, 

 and may again do so, by some optical contrivance. Now if we sup- 

 pose all the points of a plane luminous, each will give out a pencil 

 of rays, and that collection of pencils which passes through the in- 

 strument may be treated as a beam of light. In a pencil only one 

 ray passes through any point of space, unless that point be the focus, 

 in a beam, an infinite number of rays, corresponding each to some 

 point in the luminous plane, passes through any point ; and we may, 



