392 REPORTS ON THE STATE OF SCIENCE. — 1919. 



last place of decimals. The planimeter measure of the print gave '458 L 

 as its area. 



A steam indicator diagram is small ; and so, in using a planimeter, 

 considerable skill is required to obtain good results. The first of the 

 Gauss rules gives a fairly good result, seeing that only three ordinates 

 are measured ; and it may be regarded as sufficient for some practical 

 purposes. The second of the Gauss rules, with five ordinates, give as 

 accurate a result as can be obtained in the ordinary circumstances of the 

 degree of accuracy of measurements. But for the approximate value of 

 an integral, much greater accuracy is often desirable ; and so the Gauss 

 seven- ordinate rule would be more effective, seeing that it gives as 

 accurate a result as would be obtained by taking thirteen equidistant 

 measures of the subject of integration. 



It should be pointed out that a steam indicator diagram possesses 

 the smooth quality assumed by the analysis which is used in establishing 

 Gauss's rule. But a Diesel diagram often does not possess this quality ; 

 and so the rules cannot always be expected to give similarly satisfactory 

 results for Diesel diagrams. A skilful use of the planimeter is, of course, 

 effective. 



4. The preceding examples arise from the cases when the order of the 

 Legendre function is odd. A single example will suffice to indicate the 

 form when the order is even. We shall take n—A. 



The places where the ordinates are to be measured are given by the 

 roots of the equation 'Bi{y)=0; that is, if i/ = t, we have 



35^2 _30i + 3=0, 

 and therefore 



35a, ' = 1.54- (120)S 35a,2=15-(120)4. 

 Then 



a.^a,'^ 



.1 



^ -34785, 



2 3(120)^ 

 r<2" — a," 



We then have, by this rule, 



^|7(x)Ja;=iAi|/(c,)+/(c,')}+iA2{/(c,)+/(o20|, 

 where 



