THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 65 



where as are present. This is a consequence of the method of procedure explained 

 in 28. 



In calculating the contributions to c u , c ]3 , ... , I have (unwisely) taken the variables 



L' M' 

 to be I/, M', ... , instead of , -yj, ... . In virtue of relations (108) to(llO), only three 



Ct 



contributions need have been calculated, but I have calculated five out of the six quite 

 independently, so as to have two independent checks on the computations. 

 For the contribution to c n , I find 



4c n = I -7 | i 



Jo A \A 4 A 3 



-L'/J 27 I 3 I ]+w(-> T 3 - 



! ^ " A A A A ; b A A A A A 1 ~, i A A A T 1U ~T: i \ \ Tt ~TS J 



/ q o \ / 1 



4. "W'/JiT _T )-L/M__T T 



1 8c" AAC 8r 2 IAACC / V46V AAC 4? IAABC 



-^-1 -i-T X T '/ T T 5 T 



2 2 1 AAA . 2 "-AAAC 2 ^AAAC / "*" ' M ^73 J AAIi 77"-> ^AAAB 



B tt 4c6 C &1 U 4'; 





The other five contributions can be similarly written down. It then can be verified 

 algebraically that the three relations (108) to (l 10) are satisfied ; this provides a check 

 on the method and the algebra, but it hardly seems worth exhibiting it here, as the 

 numerical checks to be given later provide simultaneously checks on the method, the 

 algebra, and the computations. By independent computations I find for the contribu- 

 tions to c n , CM, Cgg, c, 2 , c l3 , the following : 



4c H = 



+ 0'01515472/'-0-0125112W-0'OOG581149/// 

 4c, 2 = 0'00044G082l/ + 0'1490178M / + 0'27801GlN' 



-0'390933lZ' + 0'00681885m'-0'00912350/i' 

 4c 33 = 



4c 12 = -0-003687295L/-0-14G5540M 7 



4c 13 = -0'004468110L' + > 0698178M'-0'4453435N / 



-0'026639GO/ / + 0'11100470m'-0'02290710?i'. 

 From these 



4(6c 11 + c 12 + c l3 ) = -0 < 000000007L'-0'0000001M' + 0'OOOOOOON / 

 -0'0000003^-0'0000004m' + 



I have not calculated c 23 independently, but a check is provided by the comparison of 

 the values of c 23 deduced from the above figures by the use of equations (109^ and (l 10) 

 VOL. ccxv. A. K 



