K. I'kahson 



13 



Hencc it, is vU':iv tliat aiiy e caii 1«' expressed iit oucu in terms of Üiu iiioments 

 and of DUO or otlier of the dettTiuinant.s 



1, l/.S, l/ö. 1/7, ... 

 1/:J, 1/5, 1/7, 1/1), ... 

 l/ö, 1/7, 1/9, I/Il,... 



Iß, l/ö, 1/7, 1/0, ... 

 1/5, 1/7, 1/9, 1/11,... 

 1/7, 1/9, 1/11, 1/18,... 



und l-lieir respective niinors. 



I( is thus (juite easy to oxpross the general result of working with e«, «i ... e,,-,. 

 But as a matter of practica! ai)plication, it wonld involve far too much troublesomc 

 arithmetie to calciilato momeiits beyond the tiftli or .sixth. We have accordiiigly 

 only to caiculate once and for all the numerical coeEficients of the X's in the; values 

 of the e's for the first few cases and these will serve for all future applications. 



Ca.se (i). To fit a straight liiie to a series of observations. 



Let the linc be y = yjeu + e, 'j 



Then Xg = e,,, X., = ^e,. 



Thus the cfjuation of the line is 



y = y« (\, + 3x, I 



GeoTnetncal Gonstriiction. Let the broken line ^jB be the observations and 

 A'B' the best straight line. Then A'B'EF and ADBEF must have the same first 

 moments and the same area. Let CK be the vertical through the centroid of the 

 observations, i.e. obtained by taking their mean FK. Now the trapezium may 

 be considered as made up of two triangles A'EF and A'B'E, the centroids of 

 which lie in the vertical lines G^Hj^ and G.,H„ trisecting FE. Hence the area 

 A'B'EF acting in CK must be equivalent to the areas A'EF and A'B'E, or 

 l X A'F and l x B'E, acting in G,H, and G„H.,. Now A'F+ B'E is known, for it 

 equals S^/o, the double of the mean Ordinate of the observations. 



Take Ol = 2yo and from any poiiit 0, draw 00 to meet G^H^ and CK in t and u. 

 Draw tiv parallel to Ol to meet G,H., in v, and then draw 02 parallel to tv to meet 

 Ol in 2. We shall then have 02 = A'F and 21 = B'E, the required lengths, 

 which fully determine the line A'B'. 



The construction given is the familiär graphical one for finding the components 

 in the lines G^H^ and G.H.. parallel to CK of a force 2^„ acting in CK. The prin- 

 ciple of moments would also give a Solution. Thus take moments about //j : 



2/ 

 A'Fx^ = -2y„xH,K, 



