114 Prof. B. Stewart and W. Dodgson. [May 29, 



10. We have found it necessary to push things not merely by the 

 multiple of a whole division — right or left each je&r — but by the 

 multiple of half a division. 



To accomplish this we must obtain for every alternate year a series 

 of half-way points, which is best done by converting the series of 

 such years into curves, and using these to give us the half-way points. 



For instance, should we wish to pull things to the left half a division 

 each year, we should adopt the following arrangement. — 



1858 . . (0|) 58 , (H) 68 , (2l) S8 . . (21|) 68 , (22|) 68 , (23J) 58 . 



1859 . . (1) 6 „ (2) 59! (3) 59 . . (22) 59 , (23) 59 , (0)„. 



1860 . . (1*)» (2i)eo- (3i)eo • • (23^), (0|) 60 . 



1861 . . (2) 61 , (3) 61) (4) n . . (28) n , (0) 01> (l) n . 



A further extension of the same method would enable us to go from 

 halves to quarters of a division, but we have not. yet met with any 

 inequality that is not sufficiently well indicated by half divisions. 



11. We have found the following plan very useful in abridging the 

 labour of these reductions. 



Sixteen perfectly similar strips of thick paper are taken, one for 

 each year. Each of these strips is divided into 48 small com- 

 partments, a vertical black line being ruled at the beginning, the 

 middle, and the end. Upon the first of these strips the inequality, 

 say for 1858, is written in duplicate, so that the second 24 

 figures are a repetition of the first. The same is done for the other 

 years. These strips are then attached to a frame which allows them 

 to slide along with regard to each other, and at the same time only 

 exposes 24 lines of figures at a time. By this method they can 

 easily be arranged according to any given order, and the sums made 

 with very little trouble. 



It is further desirable to exhibit all the positive values, say in black, 

 and all the negative values in red, so that the eye may easily distinguish 

 between them. 



The following will exhibit the nature of the arrangement — only to 

 save space we take an inequality consisting of only four terms. 



t ^ ) 









(0) 68 



(1)m 



(2)58 



(3)58 



(0)58 



(1)58 



(2) 58 



(3)58 







(0)50 



(1) 59 



(2) 59 



(3) 69 



(0) 5 9 



(1)59 



(2)59 



(3)59 







(0)« 



(!)«, 



(2) 60 



(3)(i(i 



(0) 60 



(1)60 



(2)60 



(8)« 









(1)61 



(2) 81 



(3) 61 



(0)oi 



(1)6) 



(2)61 



(8) 8 









