Pi = 



Numerical Calculation of the Jacobian Ellipsoids. 



(13) 



4. In bringing (2) into a form suitable for our purpose, the 

 same method as in the previous paper is adopted, which may be 

 briefly sketched as follows: — 



Substituting from (11) and (13), équation (2) becomes, after 

 reduction 



[^Ä"^o(2iO + &:d:ß,{9,v)^ 2r = a;äJ(^v) + âX(2v)-7râ.^â,(2v). (14) 



On replacing the ^-functions Ijy their expressions in A-series, it is 

 immediately seen that each member of the above equation is 

 divisible by /r; hence a solution of this equation is A=0, from 

 which it follows that 62=^3 or a=b. This solution determines the 

 well-known series of ]Maclaurin's spheroids, the consideration of 

 which, however, is not an aim of this paper. 



Dividing out the factor /r on both sides of (14), arranging 

 the quotients in powers of h, and transposing, we obtain 



A, + AJr + ÄJi'+ +A,h'-+ =0, (15) 



where ^4' s are known functions of -2' alone. 



If, in the last equation, we put ^ = 0, then 



A, = (4-cos2^)2,j-16sin^ + 5sin2." = 0, (16) 



where ^ = 2-i\ 



The value of z, sav 2^0, which satisfies (16) corresponds to the 

 starting point of the Jacobian from the revolutional series, and can 

 be found, as will be given below, by the method of successive 

 approximation. 



Now, Zf) being known, the general value of 0, which is defined 

 by (15) as a function of h, can be expanded in the neighbourhood 

 of A = 0, thus: — 



z = Z(, + aji- + aJi' + + aji'- + (17) 



The numerical coefficients 2^0, «1, «2, , «5 were recomputed and 



«6 was newly found for the present paper, the result being 



