METHODS OF INTEKPOLATION. 335 



J2 + 2 of the adjusted double series a miuimum, the weight of the middle 

 term must be increased from 5 to 8^, so that — 



will be the formula required. 



APPENDIX III. 



Since the present memoir was written, the author has met with a 

 small work by Schiaparelli, designed with especial reference to the 

 reduction of meteorological observations, and entitled Sul modo di rica- 

 vare la vera cspressionc delle leggi della natnra dalle curve cmpirklie; Mi- 

 lan, 18G7. That work, it is proper to acknowledge, an.ticipates to a 

 certain extent the second method of adjustment here given. It con- 

 tains, in section 45, a development of the general relation, or system of 

 conditions, wliich exists between the numerical coeflicients or weights, 

 in formulas for adjusting the middle one of any group of an odd num- 

 ber of terms in a series. The mode of demonstration is quite diflerent 

 from the one here followed, and its author does not obtain any of the 

 special adjustment formulas wliich have here been constructed and re- 

 commended, such as (17), (19), &c., (53), (54), &c., or (G9), (71), «&c. He 

 gives instead, on page 17, that special case under our formula (13) which 

 arises when we take — 



W2 = Wl, ffi = ^(Mi — 1) 



and also gives, on page 47, the formulas which render the probable error 

 of the adjusted term a minimum. We have seen that these last can be 

 derived from equations of condition by the method of least squares; that 

 their weights form series of the second order; and that the adjustments 

 which tliey make are not nearly so smooth and regular as those made by 

 formulas whose weights follow a curve which is continuous with the 

 line of the zero weights. The method of least squares presupposes that 

 the assumed algebraic equation, of a degree not higher than the third, 

 can accurately' represent the true law of the natural phenomenon 

 throughout the whole group of ternis included by the formula ; and, more- 

 over, to give full scope to the method, the number of terms included 

 ought to be large. These conditions will be but imperfectly fulfilled in 

 practice, and since the true law of the natural series is supposed to be 

 continuous and not irregular or broken, it appears probable, or at least 

 quite possible, that the system of weights which makes the smoothest 

 adjustment will also make the most accurate one. 



The method which Schiaparelli gives on pages 23 to 30 of his work, 

 for obtaining the values of the constants in empirical equations of alge- 

 braic or circular form when the arithmetical means of the terms in cer- 

 tain groups are taken as data, is not equivalent to the first method here 

 proposed. It requires for completeness two sets of formulas, one to be 



