THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 61 



others. The required integrals can all be obtained by successive applications of 

 formulfe of the following types, which can be verified without trouble : 



(a*-b 2 ) JAB = JB- JA 5 (n'-V) IAB = IB-!A, &c. 

 3 IBB = 2J B 1 BC 1 AB ; a J AA = J A 1 AA , & c - 



2 



2 n - J A n B - J A n 



A n B A n c 



(2n+ 1) I A n+i = 2 J A -IAB-IAC- 



A great number of the integrals can be calculated in two or more ways, and owing 

 to this circumstance it is possible to provide very complete checks on the accuracy of 

 calculation. Two complications- are worthy of mention. 



In the first place one must consider the ordinary cumulative error of all prolonged 

 computation. Since b 2 is nearly equal to c 3 , any error present will be increased when 



we evaluate a new quantity of the type^ ' .^ 2 > consequently where a quantity 



can be evaluated in two ways, the one in which division by b 3 c 2 is not involved has 

 been taken to be the true value, and the one derived by division by I? c 3 , has been used 

 merely as a check, and has generally been found to differ in the sixth or seventh place 

 (or near the end of the computations even in the fifth place) from the other values. 



Secondly, if the 10 integrals used as base were known with perfect accuracy, the 

 checks ought to be satisfied fully except for the error in the last one or two figures. 

 But, as has been indicated in 23, the 10 integrals are not themselves perfectly self- 

 consistent, so that different methods of computation will lead to a difference of the 

 final results comparable with the errors in the basic integrals. 



The following table gives the values I have selected as the best for the various 

 integrals required. I have not thought it necessary to record the checks or estimate 

 the probable errors here, as a much more searching test of the accuracy of the whole 

 computation can be provided at a later stage. 



J = 1'8401326. 



J A = 0'2583003, J B = 07G47290, J c = 0'9769708, 



JAA = 0'05262769, J AB = 01751040, J AC = 0'2293883, 

 J BB = 0-6516017, JBC = 0'8813026, J cc = 1 '2044842, 



JAAA = < 011873224," JAAB = 0'04234772, JAAC = 0'05641920, 

 JABB = 0'1647550, J ABC = 0'2254075, J ACC = 0'3112352, 



JBBB = 0'6830283, J BB c = 0'9537991, J BC c = 1'9011148, J C cc = r9011148; 



