[23] POND CULTURE. 439 



each two will increase by an equal space, so tliat wv, luive hero a pro- 

 gression whicli, if the number of squads is an even one (equal to the 

 terms of a progression), would increase at the same rate from squad to 

 squad, even if we had several hundred or any even number of squads. 

 In this progression the first term, A, is equal to the distance between the 

 first squad on the ditch and the portion of the dike assigued to it 

 (= -v/l«^+l t^), and the last term, U, is equal to the distance between the 

 last squad and their portion of the dike (= \/ i)a^-{- 10 b-), and the num- 

 ber of all the terms of the progression n is equal to the total number of 

 squads (in this case 10), from which results the above formula for ascer- 

 taining the sum total of the distances along- which the earth has to be 

 carried. This progression will be interrupted as soon as we take an 

 odd number of squads (equal to the terms of the progression), because, 

 as has been shown, the progression is oidy produced by adding- the dis- 

 tances of each two squads. In order to use the formula given for pro- 

 gressions when we have an odd number of squads (if we could not use 

 the calculation by progression we would have to find the distance of 

 each squad, as was done for the 10 squads ; these would have to be 

 added, which, however, would be a very tedious work if the number of 

 squads was large, and would, moreover, easily give rise to errors in the 

 calculation owing- to the long rows of figures), we must consider the 

 last but one squad as the last term of the progression ; we would then 

 have to find the sum of this progression, to which would have to be 

 added the distance of the last squad, in order to get the sum total of the 

 distances of all the squads. 



Supposing- we have an odd number of squads, say, instead of 10, 9, we 

 would get the following distances: 



1st squad = V la--\-lh^ 



2d squad = Vlli^-^TP 



3d squad = V^i¥~-^1? 



4th squad = V:i«2+4p 



5th squad = VSa^+ob'^ 



6th squad = V 3^+6P 



7th squad = VTa:^^TP 



8th squad = \/^iaF^S¥ 



9th squad = V^¥+9b'^ 



the sum of which, S, the first squad being A, and the last even one u 

 (here 8), and the final odd squad Z (here 9), and the number of even 

 squads n (here S), would be as follows: 



