314 METHODS OF INTERPOLATION, 



and consequently— 



M=A+^^ sin -^[Bsin( -^ )+Ccos( ^' )] 



[Dsin2(^)+Ecos2(^)] 

 [Fsin 3 (%-)+G cos 3 ("^)] +&c. 



N . 27:11 



-^ — sin -xr 

 2-n N 



N . 3t77? 

 "o — sin -^,- 



The expressions for S and M are tbus identical in form with the expres- 

 sion for y, the constants B and C, D and E, P and G, &c., being merely 

 multiplied, in the expression for M, by the known factors — 



/N . TTwN / N . 2nn\ / N . 3rw\ „ 



I — sin ^ ), ( -^ — sm -^r,- J, ( n — sin -^r^ i, &c. 



This property has already been discovered, and utilized in forming the 

 equations of curves representing annual variations of temperature, the 

 observed monthly means being taken as data.* (See the Edinburgh Neic 

 PhilosopMcal Journal for July, 1801, and the American Journal of Sci- 

 ences and Arts for January and September, 1863. )t The quantity M is 

 there regarded as the mean value of the infinite number of ordinates, or 

 " instantaneous temperatures," which fall within the interval «, and not 

 as the arithmetical mean of a finite number n of terms taken in a group. 

 In general, to obtain an expression for the sum S of the terms in a 

 group, it is not necessary that any integration should be performed. 

 Since the form of the function ^ is arbitrary, it follows that the form of 



/ ydx is arbitrary also, and may be assumed at pleasure. ])enoting by 

 f{x) any continuous function of one variable, let us substitute in the 

 place of the variable first iP+^ and then a--— ^, and let the difference 

 between the two results be — 



7/=/(^+i)-/(;r-i) . . . (43) 

 Let values in arithmetical progression, whose constant difference is 

 unity, be successively' assigned to x in the above expression. In the 



series formed by the resulting values of « let any group of n terms be 



. ' i . 



* For thft purposes of our preseut motliod, it will be most couveuient to write — 

 y=A+^ I B. siu (|^) + C, cos(|~^) } +| { B. sin ^Qf) 



+ C.cos2(^^^ |+'J:|B3.sin -('^-) + C3Cos3^Y') ] + ^^''^ 

 Then, after integrating, we .shall have— 



S=A« + sin (^;){B.,sin (^^) + C.oos (|^) } 



+ -.2(^)|B.sin2(f) + C.cos.(|^)} 



+ sin C^J { 1-3 sin :? ^~^'^ + C3 cos :^(~|"') } + &c. 



For other fornnilas, see Ajipendix IV. • 



t These articles are by J. 1). Everett. 



