1863.] 



of a Solid Body about a Fixed Point. 



55 



The coefficient of — 0' 



Hence (dividing throughout by V) (^0) becomes 



0H2.S0^- + (S' + ^)0 + S^-V=O; 



or, what is the same thing, 



(0 + S)'-S(O + S)^ + ^(O + S)-V=O 



or, as it may also be written, 



(11) 



A-(0 + S), 

 H, 



H, 



B-(a+S), 



F, 



G 

 F 



C-(0 + S) 



= 0. 



It will be seen by reference to (9) that the values of d determined by this 

 equation are equal to the ratios of the coefficients of the squares of the new 

 variables respectively in the equivalents of (2) and (7). The coefficients of 

 transformation are nine in number ; if therefore to the six equations of 

 condition (8) we add three more, the system will be determinate. 



Let three new conditions be 



(A...F...X««.«.)-^ = l, -] 



(A...F...X/3A/3=)^ = 1. V (12) 



(A...F...Xyy,r,X=lJ 



then the variable terms of (2) will take the form of the sum of three squares, 

 and the roots of (11) will be the coefficients of the transformed expression 

 for (7). Or, if 0, 0„ d.^he the roots of (11), (2) and (7) take the forms 



In order to determine the values of the coefficients of transformation 

 a, a^, we have from (9), 



(a-^-A0)a+(|^-H0)a4-(^-Ga)a,=O, ^ 

 (f^-H0)a+(B-^-B0)a+(df_F0)a,=:O, .... (14) 

 ((&-G0)a+(dr-F0)a+(€-^-C0)^^,=O; ^ 

 from the last two of which 



a: BC-(a3 + ^)a' + a''-(BC + CB-B + C^)+BC02 

 -dP -2drF0 -F-02 



dp 



= a : VA + H^ + (B + C^ + B33 + CC + 2Fdr)0 + ^0' 

 = a : VA + ^^ + (2V-H?^-G(&-A^+S^)0 + ^0- 

 = cc : VA + ^^ + (V + S^) + ^02 



= a:V(A + 0) + ^(^4-S0 + 0^); 

 or, writing for brevity 



^-}-S0 + 0-=T, 



