TWENTY-SIXTH ANNUAL MEETING. 33 



From these examples we may deduce the following formula: 



Let cd be the common difference of the series to be added; then, having ar- 



ranged the square series as far as desired in either direction from 0, \ — is to 



be added to 0, and — — to be added to the term in the square series equal to 



'cd^ 2 



, so as to make the sum of the two 0. The other terms of the series "to be 



I cd] '^ { cd \'^ [ cd\^ 



added will be — + cd,\ — + Scd, — + 3 cd, etc. 



{cd'\ » 

 In the even series of the series to be added falls under — of the square 



series. In the odd series in the series to be added does not occur; because, of 

 course, is not an odd number. 



To show the harmonies that exist between the arithmetical series and the 

 square series we will inspect a few of them. Take, for instance, the first example 

 given, having a common difference of 4: 



Squares 36 25 16 9 4 1 1 4 9 16 25 36 49 



Quartate series. .... -20 -16 -12 -8-40 4 8 12 16 20 24 28 32 



Sums 16 9 4 1 1 4 9 16 25 36 49 64 81 



Remainders 56 41 28 17 8 1-4-7-8-7-4 1 8 17 



Difference -40 -32 -24 -16 -8 8 16 24 32 40 48 56 64 



Common difference. 8 8 8 8888888888 



Difference of rem. . . 15 13 11 9 7 5 3 1-1-3 -5 -7 -9 

 Difference of sums.. 7 5 3 1-1-3-5-7 -9 -11 -13 -15 -17 



Differences 8 8 8 8 888 888888 



Take another example, with a common diffei-ence of 2: 



Squares 25 16 9 4 1 1 4 9 16 25 36 49 64 



Alternate series, -9-7-5-3-113579 11 13 15 17 



Sums 16 9 4 1 1 4 9 16 25 36 49 64 81 



Remainders 34 23 14 7 2-1-2-1 2 7 14 23 34 47 



Difference -18 -14 -10 -6-2 2 6 10 14 18 22 26 30 34 



Common diff 4 4 4444444 4 4 4 4 



Diff. of rem 11 9 7 5 3 1 -1 -3 -5 -7 -9 -11 -13 



Diff. of sums.... 7 5 3 1-1-3-5-7 -9 -11 -13 -15 -17 



Differences , 



PENTAGONAL SERIES. 



Add 



1 7 18 34 55 81 112 148 



X 40 = 40 280 720 1360 2200 3240 4480 5920 



4- 9 = 49 289 729 1369 2209 3249 4489 5929 Square series. 



l/ = 7 17 27 37 47 57 67 77 



Thus even the pentagonal series, under certain manipulation, is resolved into 

 the squares of a certain series regulated by the constant 5. 



HEXAGONAL SERIES. 



The method of obtaining this series is shown under " cubic series." 



