K. I*KARSON 7 



A ghiiico ;it thc aeeoinpanying Figs. 5, ü, 7 aiid 8 will show tluit thc lit has 

 been nmch improved by adopting thc systcmafcic method of momcnts. At thc 

 sarae time using a Brunsviga for all niultiplications and divisions and a Compto- 

 nictor for additions the labour is not very severe. Of course it is not contcnded 

 tbat tbis accuracy is nccessary in the present case ; Mr Galton's approximations 

 are probably close enough for tbe ends he had in view. We have only uscd his 

 data to illustratc a niothod, whicb may be of scrvico for special cascs, wherc thc 

 best avaihible deterininations of the constants are needed. 



(10) In calculating the moments in the previoiis illustration we have simply 

 concentrated along thc mid-ordinates. This was close enough for the pur|)ose of 

 illustration. When the ordinates of a curve which rcrpiires fitting are truo 

 ordinates, say for exaraple, measurements obtained by ob.servation, theii- irregu- 

 hirity is often such that it appears idle to use complex quadrature formulffi. 

 Such formulas give very good results, if the ordinates are those of a mathe- 

 matical forniida, or if we have a fairly smooth System of points. But a very 

 frequent case is a case like that of the acconipanying figure, in which a quad- 



rature formula for the moments seems idle and yet we are scarcely justified in 

 concentrating along the ordinates to find the moments, if the base unit be not 

 indefinitely small. Here it seems reasonable to take the area and moments of the 

 System of trapezia as fairly representing the area and moments of the curve to be 

 fitted. It would be idle to use a formula like Simpson's, for exaniple ; because 

 the changes in curvaturo in the curve, which would be ailowed for by Simpson's 

 method of striking a parabola through three successive points, have clearly no 

 existence in the general sweep of the observations and are due only to irregularities 

 of Observation. 



Accordingly we want an exprcssion for the moments of a systcni of trapezia. 

 Let the ordinates he y^, yi, y^... y^ and the corresponding abscissas .%, «j, x^ ■•• a;,«. 



