854 



Every term of the series is composed of (p+ i) prior terms ; also 

 if q be added to any term it will equal (g+i) prior terms. 



20. It is then proved that the property of the triangular numbers 

 is not destroyed by adding any (the same) number to each term, it 

 is merely postponed, and commences at a higher number according 

 to the magnitude of the number added. 



21. The same is proved in respect of the addition of any common 

 arithmetical series. 



22. The paper concludes by suggesting a proof that every number 

 may be composed of 4 triangular numbers, derived from the considera- 

 tion that if the triangular numbers be indexed or numbered thus — 



1 2 3 4 5 Sec. indices. . 



1 3 6 10 15&C. A'^nos. 



Any number between 2 triangular numbers can be formed by 4 

 triangular numbers, the sum of whose indices shall be not less than 

 the sum of the indices of the 4 triangular numbers that compose the 

 smaller triangular number, and not greater than the similar indices 

 of the larger; and generally (after a limited number of terms) the 

 sum of the indices of any intermediate number will be exactly the 

 sum of the indices of the smaller number : to illustrate this, all the 

 numbers between 91 and 105 (2 triangular numbers) are shown to 

 consist of 4 triangular numbers, whose indices exactly equal 25, 

 which is the sum of the indices of the 4 triangular numbers into 

 which 91 may be divided — 



6 6 6 7 

 thus 91~2i -1-21+21+28 



the sum of the indices 6 + 6 + 6 + 7=25; and every number be- 

 tween 91 and 105 may be composed of 4 triangular numbers, whose 

 indices added together will equal 25 ; but the nature of this inves- 

 tigation cannot be made intelligible without reference to the table 

 itself, which the paper contains. 



If this attempt is successful, the whole of Fermat's theorem of the 

 polygonal numbers may be proved without reference to Lagrange's 

 proof of the case of the squares (the second case) derived from the 

 properties of the prime numbers. The writer intimates an intention 

 of making further communications on the same subject. 



13. "On the Analysis of Numerical Equations." By J. R. Young, 

 Esq., Professor of Mathematics in Belfast College. Communicated 

 by Sir John W. Lubbock, Bart., F.R.S. &c. 



The object of this communication is to diminish the labour at- 

 tendant upon existing methods for the analysis of numerical equa- 

 tions. As Budan pointed out intervals, within the bounds of the 

 extreme limits of the roots of an equation, in which all search for 

 roots would be fruitless, so here the author seeks for what he terms 

 " rejective intervals" among those which Budan had retained. This 

 he proposes effecting by transforming the first member of every 

 equation X=0 into 



X={F+\/F^}x{F-VP^} (1.) 



which the author calls decomposing it into conjugate factors] in 



