of Variable Qiuditities. 5 



that next it niul next but oiu^ to it. Then tlu^ poiiiis to bo 

 substituted bis(H'i tlie intercc^pts B />, C c, &c. in V, S &c. It' the 

 curve wliich nuiy be (h'.'nvn tlii-ouoli P, S S:c. still secMii.s too 

 sinuous, repeat the operation. 



M JsJ Q V W 



The arithmetical ap[)li('ation of this })rocessis obviously very 

 simple; for 



and tlicrefore the correction to be applied to any ordinate z/^r is 

 is iA'^'i/^-i. 



We can see how it is that this proc^ess tends to improve the 

 curve. The observed or ffiven vaUies of the function consist of 

 two parts, the tirst re[)r(\s(^ntino- the j)rincl[)al (^vent or wave 

 whose law of variation is to be found, and the scH'ond the errors 

 or ripples which are to be eliminated. Now the observations 

 are supposed to be so close as not to admit of very large dif- 

 ferences between the successive values of the principal eve^it ; 

 and therefore their second differences will be small. On the 

 other hand, the errors will be some positive and some n(^ga- 

 tive ; and th(M•(^fore their second diffen^nces will be v(ut irre- 

 gular, and probably nmch liirgor on the whole than th(^ errors 

 themselves. The second ditterences of the observed values are 

 the sums formed by the addition of these two sets of second 

 differences ; and the justifiability of the process depends on 

 the assumption th;it the increjis(^ of the latter will Ix^ snffici(Mit 

 to render tlu^ diminished values of the Ibi-mer insigniticnnt by 

 comparison. We thus obtain a series of (piantities which de- 

 pend principally on the errors, except in ca«e the errors are 

 small, or in case of a run of luck in the signs and magnitude 

 of the errors, such as to make them apparently conform to law 



