( 247 ) 



we can develop tiie lotal siiiuosily of I he reduced and compensated 

 correction as follows : 



^^^ J K^it"-)'^^" J L <-it 



(P{S—f] (psfi iPS^ d-s>y <i:'St'~Y 



tie '^ lie " de " ae 



• dt. 



In order to i^ive llie grealest probahilily lo I lie sei'ies L we must 

 choose ,<•', y, z and a so, that (he partial derivalives of //, with 

 regard to each of these are zero ; i.e. that they satisfy the following 

 relation : 



ƒ[ 



d\S—f) d\S'^ <P5^ 



de 



dt' 



dt' 



d^S^ d^SP' 

 — u 



dt' 



dt' 



d'Sf> 



0, 



and 3 others, which we olitain by substituting for the last factor 



and 



— , successively 



df' ' " dt' 



Detinite integrals, such 



dt' 



rd' ^ 



as I —- 



J dt 



de 

 d'Sf^ d' s^^ 



de 



dt, which here occur as 



coefficients, can be computed in the following w^ay: 



J 



d' S^ dS^ _ 

 df '~dt ~ 



" s'-i dsn r 



de ' dt J J 



dS^ d' s^^ 

 dt = 



dt dt' 



= 2 [cJ^ ,fj - G 21^ en (Sr - S,y- = 2 [c^^ </-] - 6 :S » eH qI 



a 



If in the last term we also substitute for 



■'■S 



B li 



3 u t'„ , (cr — Cfj) , we have 



d' aS^" d' S^ „ ~ - ^^ 



-iriiF'' = 1'^ ■'' ^ + - ^» ("" - "'' 



and after interchanging the indices ^ and v> in the second member also 



1 have comi)ulcd these coeflicients according to both formulae, and 

 thus obtained a rigorons test. 



At the beginning and the end of Ihe interpolation the quantities 

 c are always zero, so that |)roducts such as r'^'y- would be zero at 

 either end. Witii a view, however, to the cojitinnation of this com- 

 pntatiou for next year, I ha\ c not closed the interpolation-compu- 



