ON gauss's theorem for quadrature, etc. 391 

 Now 



*! tt2 a3 L -1 



A.= 



ai-^(a,^-a./)(a,2-a,2)' 



a3"^(«2^-ai')(a2^-a3^) ' 



When the foregoing values of a,^, a^-, aj^ are substituted in each of the 

 quantities A,, A;, A3, and the symmetric functions of a,*, a>-, a-j^ are 

 substituted in Aq, and when reductions are effected, it is found that an 

 approximate value of the integral 



■ L. r fix)da 



7)-q] 



I 



is given by the expression 



where 



•2090/(co) + -0647{/(cO+/(c,')| 



+ -1399|/(C2)+/(C,')|+-1909|/(C3)+/(C3')}, 



C], c/ = ^2^ + q)±\{p-q)at, 

 c-2, c.2'^l{l) + q)±Up — q)a2, 

 c-i, C3'=h{2) + q)±h{p-q)a3. 



As an example, the rule was applied to a carbon print of a steam 

 indicator diagram ; and other rules were applied to other carbon prints of 

 the same diagram. The measurements were made for me by Mr. F. Lord, 

 a demonstrator in my department in the Imperial College of Science and 

 Technology. And a planimeter measurement of these prints was made 

 for me by Mr. W. E. G. Sillick, a lecturer in the same department. The 

 results were as follows : — 



The Engineers' rule of the ten mid-ordiuates, whereby the diagram 

 is divided into ten compartments of equal breadth and the mid-ordinates 

 are measured, gave •452 L square inches as the area, where L is the breadth 

 of the diagram. The planimeter measure of the same print of the 

 diagram gave "455 L as its area. 



The Weddle rule for another print of the diagram gave "445 L as the 

 area. The planimeter measure for this print gave '455 L as its area. 



• The first of the preceding Gauss rules (with only three ordinates) 

 gave '44 L as the area of a third print. The second of the preceding 

 Gauss rules (with five ordinates) gave practically -460 L as its area, the 

 last place of decimals being very slightly appreciated. The third of the 

 Gauss rules gave '460 L as its area, there being no appreciation in the 



G G 2 



